# questions

title: the-number-of-real-roots-of-the-equation-e-e-x-1-x-2-0-is
slug: the-number-of-real-roots-of-the-equation-e-e-x-1-x-2-0-is-65374
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of real roots of the equation e $\mathrm{e}^{(x-1)}+\mathrm{x}$ $-2=0$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">Let $f(x)=e^{x-1}+x-2$
<br/> <br/>check for $x=1$ Then, $\mathrm{f}(1)=\mathrm{e}^{0}+1-2=0$
<br/> <br/>So, $x=1$ is a real root of the equation $f(x)=0$ Let $x=\alpha$ be the other root such that $\alpha&gt;1$ or
<br/> <br/>$\alpha&lt;1$. Consider the interval $[1, \alpha]$ or $[\alpha, 1]$
<br/> <br/>Clearly $\mathrm{f}(1)=\mathrm{f}(\alpha)=0$
<br/> <br/>By Rolle's theorem $\mathrm{f}(\mathrm{x})=0$ has a root in $(1, \alpha)$ or in $(\alpha, 1)$. But $f^{\prime}(x)=e^{x-1}+1&gt;0,$ for all $x$. Thus
<br/> <br/>$\mathrm{f}^{\prime}(\mathrm{x}) \neq 0,$ for any $\mathrm{x} \in(1, \alpha)$ or $\mathrm{x} \in(\alpha, 1)$
<br/> <br/>which is a contradiction. Hence, $\mathrm{f}(\mathrm{x})=0$ has no real root other than 1 .</div>


title: the-number-of-real-solutions-of-the-equation-x-1-2-x-2-2-x-3-2-0-is
slug: the-number-of-real-solutions-of-the-equation-x-1-2-x-2-2-x-3-2-0-is-61077
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of real solutions of the equation
<br/> <br/>$(x-1)^{2}+(x-2)^{2}+(x-3)^{2}=0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">Equation holds if and only if $x-1=0, x-2=0$ and $x-3=0$ simultanesouly, which is not possible.</div>


title: the-number-of-solutions-of-d-x-d-y-x-1-y-1-when-y-1-2-is
slug: the-number-of-solutions-of-d-x-d-y-x-1-y-1-when-y-1-2-is-23018
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">one</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of solutions of $\frac{d y}{d x}=\frac{y+1}{x-1}$, when $y(1)=2$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">none</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">one</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">two</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">infinite</span> </li> </ul>

answer:
<div class="solution">Since, $\frac{d y}{d x}=\frac{y+1}{x-1}$<br/>$\Rightarrow \quad \frac{d y}{y+1}=\frac{d x}{x-1}$<br/>After integrating on both sides, we have<br/>$\log (y+1)=\log (x-1)-\log C$<br/>$C(y+1)=(x-1)$<br/>$C=\frac{x-1}{y+1}$<br/>If $x=1$, then $y=2$, so $C=0$ <br/>Therefore, $x-1=0$ <br/>Hence, there is only one solution.</div>


title: the-number-of-terms-in-the-expansion-of-1-5-2-x-9-1-5-2-x-9-is
slug: the-number-of-terms-in-the-expansion-of-1-5-2-x-9-1-5-2-x-9-is-48686
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of terms in the expansion of $(1+5 \sqrt{2} x)^9+(1-5 \sqrt{2} x)^9$, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>5
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>7
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>9
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>10</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>5
<br/></span> </div>

answer:
<div class="solution">We have,
<br/>$(1+5 \sqrt{2} x)^9+(1-5 \sqrt{2} x)^9$
<br/>$=2\left\{{ }^2 C_0+{ }^9 C_2(5 \sqrt{2} x)^2+\ldots+{ }^9 C_8(5 \sqrt{2} x)^8\right\}$
<br/>Clearly, it has 5 terms.</div>


title: the-number-of-times-the-digit-5-will-be-written-when-listing-the-integers-from-1-to-1000-is
slug: the-number-of-times-the-digit-5-will-be-written-when-listing-the-integers-from-1-to-1000-is-74107
tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of times the digit <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> will be written when listing the integers from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn></math>, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>271</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>272</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn></math></span> </div>

answer:
<div class="solution"><p>Any number between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>999</mn></math> is of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mi>c</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>≤</mo><mn>9</mn></math>. Let us first count the number in which <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> occurs exactly once.</p><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> can occur at one place in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>×</mo><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>3</mn></mmultiscripts><mo>×</mo><mn>9</mn><mo>×</mo><mn>9</mn><mo>=</mo><mn>243</mn></math> ways, next <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> can occur in exactly two places in <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi mathvariant="normal">C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>3</mn></mmultiscripts><mo>×</mo><mn>9</mn><mo>=</mo><mn>27</mn></math>.</p><p>Lastly, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> can occur in all three digits in only one way.</p><p>Hence, the number of times <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> occurs</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>×</mo><mn>243</mn><mo>+</mo><mn>27</mn><mo>×</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>×</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>243</mn><mo>+</mo><mn>54</mn><mo>+</mo><mn>3</mn><mo>=</mo><mn>300</mn></math></p></div>


title: the-number-of-values-of-satisfying-the-equation-39-c-3-r-1-39-c-r-2-39-c-r-2-1-39-c-3-r-is
slug: the-number-of-values-of-satisfying-the-equation-39-c-3-r-1-39-c-r-2-39-c-r-2-1-39-c-3-r-is-45446
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of values of satisfying the equation ${ }^{39} \mathrm{C}_{3 \mathrm{r}-1}-{ }^{39} \mathrm{C}_{\mathrm{r}^{2}}={ }^{39} \mathrm{C}_{\mathrm{r}^{2}-1}-{ }^{39} \mathrm{C}_{3 \mathrm{r}}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">${ }^{39} \mathrm{C}_{3 \mathrm{r}-1}-{ }^{39} \mathrm{C}_{\mathrm{r}^{2}}={ }^{39} \mathrm{C}_{\mathrm{r}^{2}-1}-{ }^{39} \mathrm{C}_{3 \mathrm{r}}$
<br/> <br/>$\Rightarrow{ }^{39} \mathrm{C}_{3 \mathrm{r}-1}+{ }^{39} \mathrm{C}_{3 \mathrm{r}}={ }^{39} \mathrm{C}_{\mathrm{r}^{2}-1}+{ }^{39} \mathrm{C}_{\mathrm{r}^{2}}$
<br/> <br/>$\Rightarrow{ }^{40} \mathrm{C}_{3 \mathrm{r}}={ }^{40} \mathrm{C}_{\mathrm{r}^{2}}$
<br/> <br/>$\Rightarrow \mathrm{r}^{2}=3 \mathrm{r}$ or $\mathrm{r}^{2}=40-3 \mathrm{r}$
<br/> <br/>$\Rightarrow \mathrm{r}=0,3$ or -8,5
<br/> <br/>3 and 5 are the values as the given equation is not defined by $r=0$ and $r=-8$. Hence, the number of values of $\mathrm{r}$ is 2 .</div>


tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
title:
the-number-of-ways-of-arranging-letters-of-the-word-havana-so-that-v-and-n-do-not-appear-together-is

slug:
the-number-of-ways-of-arranging-letters-of-the-word-havana-so-that-v-and-n-do-not-appear-together-is-27170

question:
<div class="question">The number of ways of arranging letters of the word HAVANA so that $\mathrm{V}$ and $\mathrm{N}$ do not appear together is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>40
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>60
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>80
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>100</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>80
<br/></span> </div>

answer:
<div class="solution">We can arrange the letters $\mathrm{H}, \mathrm{A}, \mathrm{A}, \mathrm{A}$ in $\frac{4 !}{3 !}=4$ ways. If one possible arrangement is
<br/>$$
<br/>X X X X
<br/>$$
<br/>Then, we can arrange $\mathrm{V}, \mathrm{N}$ at any of the two places marked with $\mathrm{O}$ in the following arrangement
<br/>$$
<br/>\mathrm{O} \times \mathrm{O} \times \mathrm{O} \times \mathrm{O} \times \mathrm{O}
<br/>$$
<br/>Thus, we can arrange $V$ and $N$ in ${ }^5 P_2=20$ ways. Thus, the number of ways in which letters can be arranged is $4 \times 20=80$</div>


title: the-numbers-3-2-sin-2-1-14-and-3-4-2-sin-2-form-first-three-terms-of-an-a-p-then-its-fifth-term-is
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
slug:
the-numbers-3-2-sin-2-1-14-and-3-4-2-sin-2-form-first-three-terms-of-an-a-p-then-its-fifth-term-is-62821

question:
<div class="question">The numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo><mn>14</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mn>4</mn><mo>-</mo><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi></mrow></msup></math> form first three terms of an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi><mo>.</mo><mi mathvariant="normal">P</mi><mo>.</mo></math>, then its fifth term is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>25</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>12</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>53</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>53</mn></math></span> </div>

answer:
<div class="solution"><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo><mn>14</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mn>4</mn><mo>-</mo><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi></mrow></msup></math> are in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AP</mi></math>. Therefore,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>14</mn><mo>=</mo><msup><mn>3</mn><mrow><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mn>3</mn><mrow><mn>4</mn><mo>-</mo><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>28</mn><mo>=</mo><mfrac><mi>a</mi><mn>3</mn></mfrac><mo>+</mo><mfrac><msup><mn>3</mn><mn>4</mn></msup><mi>a</mi></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><msup><mn>3</mn><mrow><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>84</mn><mi>a</mi><mo>+</mo><mn>243</mn><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfenced><mrow><mi>a</mi><mo>-</mo><mn>81</mn></mrow></mfenced><mfenced><mrow><mi>a</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>a</mi><mo>=</mo><mn>81</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>=</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mn>3</mn><mrow><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi></mrow></msup><mo>=</mo><msup><mn>3</mn><mn>4</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi></mrow></msup><mo>=</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi><mo>=</mo><mn>1</mn></math>  <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>∵</mo><mo> </mo><mn>2</mn><mi>sin</mi><mn>2</mn><mi>α</mi><mo>≠</mo><mn>4</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>sin</mi><mn>2</mn><mi>α</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⇒</mo><mn>2</mn><mi>α</mi><mo>=</mo><mn>30</mn><mo>°</mo><mo>  </mo><mfenced close="]" open="["><mrow><mo>∵</mo><mo> </mo><mi>sin</mi><mn>30</mn><mo>°</mo><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></mfenced></math></p><p>Thus, the first three terms of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AP</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mn>14</mn><mo>,</mo><mn>27</mn></math>.</p><p>Hence, its fifth term <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>5</mn></msub><mo>=</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mfenced><mrow><mn>5</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mi>d</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>+</mo><mn>4</mn><mo>×</mo><mn>13</mn><mo>=</mo><mn>1</mn><mo>+</mo><mn>52</mn><mo>=</mo><mn>53</mn></math></p></div>


title: the-one-which-is-least-basic-is
slug: the-one-which-is-least-basic-is-54153
question: <div class="question">The one which is least basic is</div>
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{NH}_{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NH}_{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)_{3} \mathrm{~N}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)_{2} \mathrm{NH}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)_{3} \mathrm{~N}$</span> </div>

answer:
<div class="solution">More the electron density on $\mathrm{N}$, higher will be the basicity. Density on $\mathrm{N}$ is influenced by the (i) nature of the group (+I or -I) present in alkyl group or benzene nucleus and (ii) resonance (delocalisation of the electron present on $\mathrm{N}$ ). In $\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)_{3} \mathrm{~N}$ :<br/>electron pair is delocalised to the maximum extent due to three benzene rings and hence least available for protonation, thus it will be least basic)</div>


title: the-orbital-angular-momentum-of-an-electron-is-2-s-orbital-is
slug: the-orbital-angular-momentum-of-an-electron-is-2-s-orbital-is-66667
tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">zero</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The orbital angular momentum of an electron is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">s</mi></math> orbital is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">h</mi><mrow><mn>4</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">zero</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">h</mi><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt><mo>·</mo><mfrac><mi mathvariant="normal">h</mi><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math></span> </li> </ul>

answer:
<div class="solution"><p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">s</mi></math>-subshell <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">l</mi><mo>=</mo><mn>0</mn></math></p><p>Orbital angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>mvr</mi><mo>)</mo><mo>=</mo><mfrac><mi mathvariant="normal">h</mi><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><msqrt><mi mathvariant="normal">l</mi><mo>(</mo><mi mathvariant="normal">l</mi><mo>+</mo><mn>1</mn><mo>)</mo></msqrt></math></p><p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">s</mi></math> orbital, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">l</mi></math> (azimuthal quantum number) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> orbital angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi mathvariant="normal">h</mi><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><msqrt><mn>0</mn><mo>(</mo><mn>0</mn><mo>+</mo><mn>1</mn><mo>)</mo></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi mathvariant="normal">h</mi><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><msqrt><mn>0</mn></msqrt><mo>=</mo><mn>0</mn></math></p><p>Thus, the orbital angular momentum of an electron in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">s</mi></math> -orbital is Zero.</p></div>


title: the-order-of-acidic-strength-of-boron-trihalides-1
slug: the-order-of-acidic-strength-of-boron-trihalides-1-52210
question: <div class="question">The order of acidic strength of boron trihalides</div>
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'JEE Main']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>BF</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BCl</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BBr</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BI</mi><mn>3</mn></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>BI</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BBr</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BCl</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BF</mi><mn>3</mn></msub></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>BCl</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BBr</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BI</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BF</mi><mn>3</mn></msub></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>BBr</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BCl</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BF</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BI</mi><mn>3</mn></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>BF</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BCl</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BBr</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BI</mi><mn>3</mn></msub></math></span> </div>

answer:
<div class="solution"><p>The strength of acidic character of boron trihalides depends upon <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pπ</mi><mo>-</mo><mi>pπ</mi></math> back bonding.</p><p>In boron trihalides, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pπ</mi><mo>-</mo><mi>pπ</mi></math> -back bonding occurs due to empty orbital of boron and filled orbitals of halogens.</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/bf2f12fc-7efd-4bd8-b002-c137c0c9320c-image.png" style="width: 398px; height: 255px;"/></p><p>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>pπ</mi><mo>-</mo><mi>pπ</mi></math> back bonding is shown maximum by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>BF</mi><mn>3</mn></msub></math>, as the size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">F</mi></math> are small and comparatively same. Due to this effect tendency of accepting lone-pair of electrons of boron decreases as size of halogen increases. The order of size of halogens are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">F</mi><mo>&lt;</mo><mi>Cl</mi><mo>&lt;</mo><mi>Br</mi><mo>&lt;</mo><mi mathvariant="normal">I</mi></math>. Thus, acidic nature is in order</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>BF</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BCl</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BBr</mi><mn>3</mn></msub><mo>&lt;</mo><msub><mi>BI</mi><mn>3</mn></msub></math></p></div>


tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
title:
the-output-of-an-or-gate-is-connected-to-both-the-inputs-of-a-nand-gate-the-combination-will-serve-as-a

slug:
the-output-of-an-or-gate-is-connected-to-both-the-inputs-of-a-nand-gate-the-combination-will-serve-as-a-74951

question:
<div class="question">The output of an OR gate is connected to both the inputs of a NAND gate. The combination will serve as a:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">NOT gate</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">NOR gate</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">AND gate</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">OR gate</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">NOR gate</span> </div>

answer:
<div class="solution">$(\overline{A+B})=$ NOR gate When both inputs of NAND gate are connected, it behaves as NOT gate $\mathrm{OR}+\mathrm{NOT}=\mathrm{NOR}$</div>


title: the-oxidation-state-of-sulphur-in-na-2-s-4-o-6-is
slug: the-oxidation-state-of-sulphur-in-na-2-s-4-o-6-is-40822
tags: ['Chemistry', 'Redox Reactions', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
question:
<div class="question">The oxidation state of sulphur in $\mathrm{Na}_{2} \mathrm{~S}_{4} \mathrm{O}_{6}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{+3}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{+5}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-2</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{+5}{2}$</span> </div>

answer:
<div class="solution">$\mathrm{Na}_{2} \mathrm{~S}_{4} \mathrm{O}_{6}$
<br/> <br/>$2+4 x-12=0$
<br/> <br/>$4 x-10=0$
<br/> <br/>$x=\frac{10}{4}=\frac{+5}{2}$
<br/> <br/>Oxidation state of $\mathrm{S}$ is $=\frac{+5}{2}$</div>


title: the-oxidation-states-of-sulphur-in-the-anions-so-3-2-s-2-o-4-2-and-s-2-o-6-2-follow-the-order-1
tags: ['Chemistry', 'Redox Reactions', 'JEE Main']
source_db: MarksBatch2_P3.db
slug:
the-oxidation-states-of-sulphur-in-the-anions-so-3-2-s-2-o-4-2-and-s-2-o-6-2-follow-the-order-1-87277

question:
<div class="question">The oxidation states of sulphur in the anions $\mathrm{SO}_{3}^{2-}, \mathrm{S}_{2} \mathrm{O}_{4}^{2-}$ and $\mathrm{S}_{2} \mathrm{O}_{6}^{2-}$ follow the order</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{SO}_{3}^{2-}&lt;\mathrm{S}_{2} \mathrm{O}_{4}^{2-}&lt;\mathrm{S}_{2} \mathrm{O}_{6}^{2-}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{S}_{2} \mathrm{O}_{4}^{2-}&lt;\mathrm{S}_{2} \mathrm{O}_{6}^{2-}&lt;\mathrm{SO}_{3}^{2-}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{S}_{2} \mathrm{O}_{6}^{2-}&lt;\mathrm{S}_{2} \mathrm{O}_{4}^{2-}&lt;\mathrm{SO}_{3}^{2-}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{S}_{2} \mathrm{O}_{4}^{2-}&lt;\mathrm{SO}_{3}^{2-}&lt;\mathrm{S}_{2} \mathrm{O}_{6}^{2-}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{S}_{2} \mathrm{O}_{4}^{2-}&lt;\mathrm{SO}_{3}^{2-}&lt;\mathrm{S}_{2} \mathrm{O}_{6}^{2-}$</span> </div>

answer:
<div class="solution">According to chemical bond method:<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/CiGhmbDxmHV-6qt6fbLHgAigU8yCStERw-2U_gLiYs0.original.fullsize.png"/></div>


title: the-p-p-p-angle-in-p-4-molecule-and-s-s-s-angle-in-s-8-molecule-isin-degree-respectively
slug: the-p-p-p-angle-in-p-4-molecule-and-s-s-s-angle-in-s-8-molecule-isin-degree-respectively-30445
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
question:
<div class="question">The $\mathrm{P}-\mathrm{P}-\mathrm{P}$ angle in $\mathrm{P}_{4}$ molecule and $\mathrm{S}-\mathrm{S}-\mathrm{S}$
<br/> <br/>angle in $\mathrm{S}_{8}$ molecule is(in degree) respectively</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$60^{\circ}, 107^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$107^{\circ}, 60^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$40^{\circ}, 60^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$60^{\circ}, 40^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$60^{\circ}, 107^{\circ}$</span> </div>

answer:
<div class="solution">In $P_{4}$ molecule, the four sp $^{3}$ -hybridised phosphorous atoms lie at the corners of a regular tetrahedron with $\angle \mathrm{PPP}=60^{\circ}$.
<br/> <br/>In $\mathrm{S}_{8}$ molecule S-S-S angle is $107^{\circ}$ rings.
<br/> <br/>In $\mathrm{S}_{8}$ molecule $\mathrm{S}-\mathrm{S}-\mathrm{S}$ angle is $107^{\circ} \mathrm{rings} .$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/bCMNUWQbSz_47wNXjl6sKLPpZSFLyWgnN3wewtcz7tY.original.fullsize.png"/></div>


title: the-pair-in-which-both-species-have-same-magnetic-moment-spin-onlyvalue-is-1
slug: the-pair-in-which-both-species-have-same-magnetic-moment-spin-onlyvalue-is-1-68009
tags: ['Chemistry', 'Coordination Compounds', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question">The pair in which both species have same magnetic moment (spin onlyvalue) is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+},\left[\mathrm{CoCI}_{4}\right]^{2-}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}_{6}\right)^{2+},\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}\right.$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{Mn}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right)^{2+},\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{CoCl}_{4}\right)^{2-},\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}_{6}\right)^{2+},\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}\right.$</span> </div>

answer:
<div class="solution">$\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+} \mathrm{Cr}$ is in $\mathrm{Cr}^{2+}$ form
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/96z0AGm1c1kEuq8J6tQkVtXLnCLkVRLwbk_6e_pYPtE.original.fullsize.png"/><br/> <br/> <br/>In $\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)\right]^{2+} \mathrm{Fe}^{2+}$ form. Both will have 4
<br/> <br/>unpaired electrons.</div>


title: the-parabola-having-its-focus-at-32-and-directrix-along-the-y-axis-has-its-vertex-at-1
slug: the-parabola-having-its-focus-at-32-and-directrix-along-the-y-axis-has-its-vertex-at-1-41384
tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question">The parabola having its focus at (3,2) and directrix along the $y$ -axis has its vertex at</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(2,2)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(\frac{3}{2}, 2\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{1}{2}, 2\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\frac{2}{3}, 2\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{3}{2}, 2\right)$</span> </div>

answer:
<div class="solution">Vertex of the parabola is a point which lies on the axis of the parabola, which is a line $\perp$ to the directrix through the focus, i.e., $\mathrm{y}=2$ and equidistant from the focus and directrix $\mathrm{x}=0$, so that the vertex is $\left(\frac{3}{2}, 2\right)$.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/PkW1gn8WS5cRe03gXSSNlVFkujUv8OduAh8NQ1i13KU.original.fullsize.png"/></div>


tags: ['Chemistry', 'Ionic Equilibrium', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.556</span> </div>
source_db: MarksBatch2_P3.db
title:
the-percentage-hydrolysis-of-015-m-solution-of-ammonium-acetate-k-a-for-ch-3-cooh-is-18-1-0-5-and-k-b-for-nh-3-is-18-1-0-5

slug:
the-percentage-hydrolysis-of-015-m-solution-of-ammonium-acetate-k-a-for-ch-3-cooh-is-18-1-0-5-and-k-b-for-nh-3-is-18-1-0-5-35254

question:
<div class="question">The percentage hydrolysis of $0.15 \mathrm{M}$ solution of ammonium acetate, $\mathrm{K}_{\mathrm{a}}$ for $\mathrm{CH}_{3} \mathrm{COOH}$ is $1.8 \times 10^{-5}$
<br/> <br/>and $\mathrm{K}_{\mathrm{b}}$ for $\mathrm{NH}_{3}$ is $1.8 \times 10^{-5}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.556</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4.72</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">9.38</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5.56</span> </li> </ul>

answer:
<div class="solution">$\alpha=\sqrt{\frac{\mathrm{K}_{\mathrm{w}}}{\mathrm{K}_{\mathrm{a}} \times \mathrm{K}_{\mathrm{b}}}}=\sqrt{\frac{1 \times 10^{-14}}{1.8 \times 10^{-5} \times 1.8 \times 10^{-5}}}=0.55$</div>


title: the-period-of-tan-3-is
slug: the-period-of-tan-3-is-98697
question: <div class="question">The period of $\tan 3 \theta$ is</div>
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \pi 4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pi / 2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\tan \theta$ is of period $\pi$ so that $\tan 3 \theta$ is of period $\pi / 3$</div>


title: the-perpendicular-distance-of-p123-from-the-line-fracx63fracy72fracz72-is
slug: the-perpendicular-distance-of-p123-from-the-line-fracx63fracy72fracz72-is-42619
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The perpendicular distance of \(P(1,2,3)\) from the line \(\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">The point \(\mathrm{A}(6,7,7)\) is on the line. Let the perpendicular from \(\mathrm{P}\) meet the line in \(\mathrm{L}\). Then<br/>\(\mathrm{AP}^2=(6-1)^2+(7-2)^2+(7-3)^2=66\)<br/>Also \(\mathrm{AL}=\) projection of AP on line<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/HO8MjbUXtl1LsNl2QTeKVPzlF5PrqFegm-npoYcSQXQ.original.fullsize.png"/><br/><br/>\(\begin{aligned}<br/>&amp; \left(\text {actual d.c.'s } \frac{3}{\sqrt{17}}, \frac{2}{\sqrt{17}}, \frac{-2}{\sqrt{17}}\right) \\<br/>\Rightarrow &amp; (6-1) \cdot \frac{3}{\sqrt{17}}+(7-2) \cdot \frac{2}{\sqrt{17}}+(7-3) \frac{-2}{\sqrt{17}} \\<br/>&amp; =\sqrt{17}<br/>\end{aligned}\)<br/>\(\therefore \perp\) distance \(\mathrm{d}\) of \(\mathrm{P}\) from the line is given by<br/>\(\mathrm{d}^2=\mathrm{AP}^2-\mathrm{AL}^2=66-17=49 \text { so that } \mathrm{d}=7\)</div>


title: the-ph-of-a-solution-is-increased-from-3-to-6-its-h-ion-concentration-will-be
slug: the-ph-of-a-solution-is-increased-from-3-to-6-its-h-ion-concentration-will-be-91387
tags: ['Chemistry', 'Ionic Equilibrium', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
question:
<div class="question">The pH of a solution is increased from 3 to 6 ; its $\mathrm{H}^{+}$ ion concentration will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">reduced to half</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">doubled</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">reduced by 1000 times</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">increased by 1000 times</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">reduced by 1000 times</span> </div>

answer:
<div class="solution">$\mathrm{pH}=3 \ldots\left[\mathrm{H}^{+}\right]=10^{-3} ; \quad \mathrm{pH}=6$<br/>$\therefore\left[\mathrm{H}^{+}\right]=10^{-6}$. Hence $\left[\mathrm{H}^{+}\right]$ is reduced by<br/>$10^{-3}$ times.</div>


title: the-photochemical-smog-does-not-generally-contain-1
slug: the-photochemical-smog-does-not-generally-contain-1-63403
question: <div class="question">The photochemical smog does not generally contain:</div>
tags: ['Chemistry', 'Environmental Chemistry', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{NO}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{NO}_2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{SO}_2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{HCHO}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{SO}_2$</span> </div>

answer:
<div class="solution">Photochemical smog contains nitrogen dioxide $\left(\mathrm{NO}_2\right)$, Ozone $\left(\mathrm{O}_3\right)$, PAN (peroxyacetylnitrate), and compounds containing $-\mathrm{CHO}$ group.</div>


tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
title:
the-photoelectric-threshold-of-metal-is-2000-a-the-energy-of-the-electrons-ejected-from-the-surface-by-ultraviolet-light-of-wavelength-1500-a-is

slug:
the-photoelectric-threshold-of-metal-is-2000-a-the-energy-of-the-electrons-ejected-from-the-surface-by-ultraviolet-light-of-wavelength-1500-a-is-23242

question:
<div class="question">The photoelectric threshold of metal is $2000 \AA$. The energy of the electrons ejected from the surface by ultraviolet light of wavelength $1500 \AA$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2.0 \mathrm{eV}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1.5 \mathrm{eV}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$15 \mathrm{eV}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$150 \mathrm{eV}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2.0 \mathrm{eV}$</span> </div>

answer:
<div class="solution">$\mathrm{E}_{\mathrm{k}}=\frac{\mathrm{hc}}{\mathrm{e}}\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right)=2.0 \mathrm{eV}$</div>


title: the-poh-value-of-a-solution-whose-hydroxide-ion-concentration-is-62-1-0-9-mol-litre-is
slug: the-poh-value-of-a-solution-whose-hydroxide-ion-concentration-is-62-1-0-9-mol-litre-is-99830
tags: ['Chemistry', 'Ionic Equilibrium', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8.21</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The pOH value of a solution whose hydroxide ion concentration is $6.2 \times 10^{-9} \mathrm{~mol} /$ litre is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">8.21</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">6.21</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">7.75</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">7.21</span> </li> </ul>

answer:
<div class="solution">$-\log (\mathrm{OH})=\mathrm{pOH} ;-\log 6.2 \times 10^{-9}=\mathrm{pOH}$
<br/>$\therefore \mathrm{pOH}=8.21$</div>


tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
title:
the-point-p-1-p-where-x-is-the-greatest-integer-function-lying-inside-the-region-bounded-by-the-circle-x-2-y-2-2-x-15-0-and-x-2-y-2-2-x-7-0-then

slug:
the-point-p-1-p-where-x-is-the-greatest-integer-function-lying-inside-the-region-bounded-by-the-circle-x-2-y-2-2-x-15-0-and-x-2-y-2-2-x-7-0-then-96520

question:
<div class="question">The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>[</mo><mi>P</mi><mo>+</mo><mn>1</mn><mo>]</mo><mo>,</mo><mo>[</mo><mi>P</mi><mo>]</mo><mo>)</mo></math> (where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>x</mi><mo>]</mo></math> is the greatest integer function) lying inside the region bounded by the circle <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>7</mn><mo>=</mo><mn>0</mn></math>, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>∈</mo><mo>[</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>-</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>∈</mo><mo>[</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>∪</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∪</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>∈</mo><mo>(</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>P</mi><mo>+</mo><mn>1</mn><mo>]</mo><mo>=</mo><mo>[</mo><mi>P</mi><mo>]</mo><mo>+</mo><mn>1</mn></math>, Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mi>P</mi><mo>]</mo><mo>=</mo><mi>n</mi></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is integer</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo>(</mo><mo>[</mo><mi>P</mi><mo>+</mo><mn>1</mn><mo>]</mo><mo>,</mo><mo>[</mo><mi>P</mi><mo>]</mo><mo>)</mo><mo>=</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>)</mo></math> lie inside the region of circles</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>1</mn></msub><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn><mo>,</mo><msub><mi>C</mi><mn>1</mn></msub><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><msub><mi>r</mi><mn>1</mn></msub><mo>=</mo><mn>4</mn></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>2</mn></msub><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>7</mn><mo>=</mo><mn>0</mn><mo>,</mo><msub><mi>C</mi><mn>2</mn></msub><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><msub><mi>r</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math></p><p>Both circles are concentric. </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>-</mo><mn>7</mn><mo>&gt;</mo><mn>0</mn></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>-</mo><mn>15</mn><mo>&lt;</mo><mn>0</mn><mo>⇒</mo><mn>4</mn><mo>&lt;</mo><msup><mi>n</mi><mn>2</mn></msup><mo>&lt;</mo><mn>8</mn></math></p><p>Which is not possible for any integer.</p></div>


title: the-points-0-7-10-1-6-6-and-4-9-6-form
slug: the-points-0-7-10-1-6-6-and-4-9-6-form-93202
question: <div class="question">The points $(0,7,10),(-1,6,6)$ and $(-4,9,6)$ form</div>
tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">a right angled isosceles triangle</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a scalene triangle</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a right angled triangle</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">an equilateral triangle</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a right angled isosceles triangle</span> </div>

answer:
<div class="solution">Let $\mathrm{P}(0,7,10), \mathrm{Q}(-1,6,6)$ and $\mathrm{R}(-4,9,6)$ be the vertices of a triangle Here, $\mathrm{PQ}=\sqrt{1+1+16}=3 \sqrt{2}$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\mathrm{QR}=\sqrt{9+9+0}=3 \sqrt{2} \\
<br/> <br/>\mathrm{PR}=\sqrt{16+4+16}=6 \\
<br/> <br/>\text { Now, } \mathrm{PQ}^{2}+\mathrm{QR}^{2}=(3 \sqrt{2})^{2}+(3 \sqrt{2})^{2}=36=(\mathrm{PR})^{2}
<br/> <br/>\end{array}$
<br/> <br/>Therefore, $\triangle \mathrm{PQR}$ is a right angled triangle at $\mathrm{Q}$. Also, $\mathrm{OQ}=\mathrm{QR}$. Hence, $\triangle \mathrm{PQR}$ is a right angled isosceles triangle.</div>


title: the-points-at-which-the-tangent-passes-through-the-origin-for-the-curve-y-4-x-3-2-x-5-are
slug: the-points-at-which-the-tangent-passes-through-the-origin-for-the-curve-y-4-x-3-2-x-5-are-59524
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
question:
<div class="question">The points at which the tangent passes through the origin for the curve $y=4 x^{3}-2 x^{5}$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(0,0),(2,1)$ and $(-1,-2)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(0,0),(2,1)$ and $(-2,-1)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(2,0),(2,1)$ and $(-3,1)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$(0,0),(1,2)$ and $(-1,-2)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(0,0),(1,2)$ and $(-1,-2)$</span> </div>

answer:
<div class="solution">The equation of the given curve is
<br/> <br/>$\begin{array}{l}
<br/> <br/>y=4 x^{3}-2 x^{5} \\
<br/> <br/>\frac{\mathrm{dy}}{\mathrm{dx}}=12 x^{2}-10 x^{4}
<br/> <br/>\end{array}$
<br/> <br/>Therefore, the slope of the tangent at point $(x, y)$ is $12 x^{2}-10 x^{4}$. The equation of the tangent at $(x, y)$ is given by
<br/> <br/>$\mathrm{Y}-y=\left(12 x^{2}-10 x^{4}\right)(\mathrm{X}-x) \ldots \text { (i) }$
<br/> <br/>When, the tangent passes through the origin $(0,0)$, then $\mathrm{X}=\mathrm{Y}=0$ Therefore, eq. (i) reduce to
<br/> <br/>$\begin{array}{l}
<br/> <br/>-y=\left(12 x^{2}-10 x^{4}\right)(-x) \\
<br/> <br/>\Rightarrow y=12 x^{3}-10 x^{5}
<br/> <br/>\end{array}$
<br/> <br/>Also, we have $y=4 x^{3}-2 x^{5}$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\therefore 12 x^{3}-10 x^{5} \\
<br/> <br/>=4 x^{3}-2 x^{5} \\
<br/> <br/>\Rightarrow 8 x^{5}-8 x^{3}=0 \\
<br/> <br/>\Rightarrow x^{5}-x^{3}=0 \\
<br/> <br/>\Rightarrow x^{3}\left(x^{2}-1\right)=0 \\
<br/> <br/>\Rightarrow x=0, \pm 1
<br/> <br/>\end{array}$
<br/> <br/>When, $x=0$
<br/> <br/>$y=4(0)^{3}-2(0)^{5}=0$
<br/> <br/>When, $x=1$,
<br/> <br/>$y=4(1)^{3}-2(1)^{5}=2$
<br/> <br/>When, $x=-1$,
<br/> <br/>$y=4(-1)^{3}-2(-1)^{5}=-2$
<br/> <br/>Hence, the require points are $(0,0),(1,2)$ and $(-1,-2)$.</div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
title:
the-position-of-a-projectile-launched-from-the-origin-at-t-0-is-given-by-r-40-i-50-j-m-at-t-2-s-if-the-projectile-was-launched-at-an-angle-from-the-ho

slug:
the-position-of-a-projectile-launched-from-the-origin-at-t-0-is-given-by-r-40-i-50-j-m-at-t-2-s-if-the-projectile-was-launched-at-an-angle-from-the-ho-91562

question:
<div class="question">The position of a projectile launched from the origin at $t=0$ is given by $\vec{r}=(40 \hat{i}+50 \hat{j}) \mathrm{m}$ at $t=$ $2 \mathrm{~s}$. If the projectile was launched at an angle $\theta$ from the horizontal, then $\theta$ is $\left(\right.$ take $\left.g=10 \mathrm{~ms}^{-2}\right)$\</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\tan ^{-1} \frac{2}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\tan ^{-1} \frac{3}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\tan ^{-1} \frac{7}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\tan ^{-1} \frac{4}{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan ^{-1} \frac{7}{4}$</span> </div>

answer:
<div class="solution">From question, Horizontal velocity (initial),
<br/> <br/>$$
<br/> <br/>u_{x}=\frac{40}{2}=20 \mathrm{~m} / \mathrm{s}
<br/> <br/>$$
<br/> <br/>Vertical velocity (initial), $50=u_{y} t+\frac{1}{2} g t^{2}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow u_{y} \times 2+\frac{1}{2}(-10) \times 4 \\
<br/> <br/>\text { or, } \quad 50=2 u_{y}-20
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>or, $u_{y}=\frac{70}{2}=35 \mathrm{~m} / \mathrm{s}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\therefore \quad \tan \theta=\frac{u_{y}}{u_{x}}=\frac{35}{20}=\frac{7}{4} \\
<br/> <br/>\Rightarrow \quad \text { Angle } \theta=\tan ^{-1} \frac{7}{4}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Physics', 'Motion In One Dimension', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
title:
the-position-x-of-a-particle-varies-with-time-t-as-x-a-t-2-b-t-3-the-acceleration-at-time-t-of-the-particle-will-be-equal-to-zero-what-is-the-value-of

slug:
the-position-x-of-a-particle-varies-with-time-t-as-x-a-t-2-b-t-3-the-acceleration-at-time-t-of-the-particle-will-be-equal-to-zero-what-is-the-value-of-53748

question:
<div class="question">The position $\mathrm{x}$ of a particle varies with time $(\mathrm{t})$ as $\mathrm{x}=\mathrm{A} \mathrm{t}^{2}-\mathrm{B} \mathrm{t}^{3}$. The acceleration at time $\mathrm{t}$ of the particle will be equal to zero. What is the value of $t ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2 \mathrm{~A}}{3 \mathrm{~B}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{A}}{\mathrm{B}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{A}{3 B}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">zero</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{A}{3 B}$</span> </div>

answer:
<div class="solution">Given that $x=A t^{2}-B t^{3}$<br/>$$<br/>\therefore \text { velocity }=\frac{\mathrm{dx}}{\mathrm{dt}}=2 \mathrm{~A} \mathrm{t}-3 \mathrm{~B} \mathrm{t}^{2}<br/>$$<br/>and acceleration <br/>$$ <br/>=\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)=2 \mathrm{~A}-6 \mathrm{Bt}<br/>$$<br/>For acceleration to be zero $2 \mathrm{~A}-6 \mathrm{Bt}=0$.<br/>$$<br/>\therefore \quad \mathrm{t}=\frac{2 \mathrm{~A}}{6 \mathrm{~B}}=\frac{\mathrm{A}}{3 \mathrm{~B}}<br/>$$</div>


tags: ['Physics', 'Work Power Energy', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
title:
the-potential-energy-for-a-force-field-f-is-given-by-u-x-y-cos-x-y-the-force-acting-on-a-particle-at-position-given-by-coordinates-0-4-is

slug:
the-potential-energy-for-a-force-field-f-is-given-by-u-x-y-cos-x-y-the-force-acting-on-a-particle-at-position-given-by-coordinates-0-4-is-16267

question:
<div class="question">The potential energy for a force field $\overrightarrow{\mathrm{F}}$ is given by $U(x, y)=\cos (x+y)$. The force acting on a particle at position given by coordinates $(0, \pi / 4)$ is $-$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{\sqrt{2}}(\hat{\mathrm{i}}+\hat{\mathrm{j}})$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{1}{2} \hat{\mathrm{i}}+\frac{\sqrt{3}}{2} \hat{\mathrm{j}}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\frac{1}{2} \hat{\mathrm{i}}-\frac{\sqrt{3}}{2} \hat{\mathrm{j}}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$</span> </div>

answer:
<div class="solution">$\quad \mathrm{F}_{\mathrm{x}}=-\frac{\partial \mathrm{U}}{\partial \mathrm{x}}=\sin (\mathrm{x}+\mathrm{y})$<br/>$$<br/>\begin{array}{l}<br/>\mathrm{F}_{\mathrm{y}}=-\frac{\partial \mathrm{U}}{\partial \mathrm{x}}=\sin (\mathrm{x}+\mathrm{y}) \\<br/>\left.\mathrm{F}_{\mathrm{x}}=\sin (\mathrm{x}+\mathrm{y})\right]_{(0, \pi / 4)}=\frac{1}{\sqrt{2}} \\<br/>\left.\mathrm{~F}_{\mathrm{y}}=\sin (\mathrm{x}+\mathrm{y})\right]_{(0, \pi / 4)}=\frac{1}{\sqrt{2}} \\<br/>\therefore \mathrm{F}=\frac{1}{\sqrt{2}}[\hat{\mathrm{i}}+\hat{\mathrm{j}}]<br/>\end{array}<br/>$$</div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
title:
the-potential-energy-of-a-satellite-of-mass-m-and-revolving-at-a-height-r-e-above-the-surface-of-earth-where-r-e-radius-of-earth-is

slug:
the-potential-energy-of-a-satellite-of-mass-m-and-revolving-at-a-height-r-e-above-the-surface-of-earth-where-r-e-radius-of-earth-is-78971

question:
<div class="question">The potential energy of a satellite of mass $\mathrm{m}$ and revolving at a height $\mathrm{R}_{\mathrm{e}}$ above the surface of earth where $\mathrm{R}_{\mathrm{e}}=$ radius of earth, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\mathrm{m} \mathrm{g} \mathrm{R}_{\mathrm{e}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{-\mathrm{mg} \mathrm{R}_{\mathrm{e}}}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{-\mathrm{mg} \mathrm{R}_{\mathrm{e}}}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{-\mathrm{mg} \mathrm{R}_{\mathrm{e}}}{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{-\mathrm{mg} \mathrm{R}_{\mathrm{e}}}{2}$</span> </div>

answer:
<div class="solution">At a height h above the surface of earth the gravitational potential energy of the particle of mass m is $\mathrm{U}_{\mathrm{h}}=-\frac{\mathrm{GM}_{\mathrm{e}} \mathrm{m}}{\mathrm{R}_{\mathrm{e}}+\mathrm{h}}$
<br/> <br/>Where $M_{e} \&amp; R_{e}$ are the mass $\&amp;$ radius of earth respectively. In this question, since $\mathrm{h}=\mathrm{R}_{\mathrm{e}}$ 
<br/> <br/>So $U_{h-R_{e}}=-\frac{G M_{e} m}{2 R_{e}}=\frac{-m g R_{e}}{2}$</div>


title: the-power-dissipated-in-the-circuit-shown-in-the-figure-is-30-watts-the-value-of-r-is-2
slug: the-power-dissipated-in-the-circuit-shown-in-the-figure-is-30-watts-the-value-of-r-is-2-92260
tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question">The power dissipated in the circuit shown in the figure is 30 Watts. The value of $R$ is<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/XUq1uouLQCsNcHbD4ax5-fvjbBx1QkLi8zTLXOamjXw.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20 \Omega$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$15 \Omega$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$10 \Omega$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$30 \Omega$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$10 \Omega$</span> </div>

answer:
<div class="solution">$P=\frac{V^{2}}{R_{\mathrm{eq}}}$<br/>$\frac{1}{R_{\mathrm{eq}}}=\frac{1}{R}+\frac{1}{5}=\frac{5+R}{5 R} R_{\mathrm{eq}}=\left(\frac{5 R}{5+R}\right) P=30 \mathrm{~W}$<br/>Substituting the values in equation (i)<br/>$$<br/>30=\frac{(10)^{2}}{\left(\frac{5 R}{5+R}\right)} \Rightarrow R=10 \Omega<br/>$$</div>


title: the-prefix-1-0-18-is
slug: the-prefix-1-0-18-is-22158
question: <div class="question">The prefix $10^{18}$ is</div>
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">exa</span> </div>
answer: <div class="solution">$\mathrm{Exa}=10^{18}$</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">giga</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">kilo</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">exa</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">nano</span> </li> </ul>


title: the-preparation-of-ethyl-acetoacetate-involves
slug: the-preparation-of-ethyl-acetoacetate-involves-28171
question: <div class="question">The preparation of ethyl acetoacetate involves:</div>
tags: ['Chemistry', 'Aldehydes and Ketones', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Wittig reaction</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Cannizzaro's reaction</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Reformatsky reaction</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Claisen condensation.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Claisen condensation.</span> </div>

answer:
<div class="solution">In Claisen condensation intermolecular condensation of esters containing $\alpha$ -hydrogen atom in presence of strong base produce $\beta$ -keto ester.
<br/>$\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}+\mathrm{H.CH}_{2} \cdot \mathrm{CO} . \mathrm{OC}_{2} \mathrm{H}_{5}$
<br/>Ethyl acetate
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/kguimHKvxXY6EUrB4tlSP48iJgCQkWEthvREmlpootU.original.fullsize.png"/></div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
title:
the-pressure-versus-temperature-graph-of-an-ideal-gas-is-shown-in-the-figure-below-if-the-density-of-gas-at-point-a-is-0-then-the-density-of-the-gas-a

slug:
the-pressure-versus-temperature-graph-of-an-ideal-gas-is-shown-in-the-figure-below-if-the-density-of-gas-at-point-a-is-0-then-the-density-of-the-gas-a-61039

question:
<div class="question"><p>The pressure versus temperature graph of an ideal gas is shown in the figure below. If the density of gas at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mn>0</mn></msub></math>, then the density of the gas at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> will be</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/a38ea51e-bfc8-4676-a99c-366ff646d4c0-image.png" style="width: 313px; height: 238px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi>ρ</mi><mn>0</mn></msub></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac><msub><mi>ρ</mi><mn>0</mn></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>2</mn></mfrac><msub><mi>ρ</mi><mn>0</mn></msub></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>ρ</mi><mn>0</mn></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac><msub><mi>ρ</mi><mn>0</mn></msub></math></span> </div>

answer:
<div class="solution"><p>Given, density of gas at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><msub><mi>ρ</mi><mi>A</mi></msub><mo>=</mo><msub><mi>ρ</mi><mn>0</mn></msub></math></p><p>General equation for an ideal gas</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>V</mi><mo>=</mo><mi>n</mi><mi>R</mi><mi>T</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>p</mi><mi>V</mi><mo>=</mo><mfrac><mi>m</mi><mi>M</mi></mfrac><mi>R</mi><mi>T</mi></math>  [<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>→</mo></math>mass, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>→</mo></math> molecular mass]</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>p</mi><mo>=</mo><mfrac><mi>m</mi><mi>V</mi></mfrac><mo>·</mo><mfrac><mi>R</mi><mi>M</mi></mfrac><mo>·</mo><mi>T</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>p</mi><mo>=</mo><mi>ρ</mi><mfenced><mfrac><mi>R</mi><mi>M</mi></mfrac></mfenced><mi>T</mi><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math>    [where density <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi><mo>=</mo><mfrac><mi>m</mi><mi>V</mi></mfrac></math>]</p><p>Applying Equation. (i) at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, respectively.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><msub><mi>ρ</mi><mi>A</mi></msub><mfenced><mfrac><mi>R</mi><mi>M</mi></mfrac></mfenced><msub><mi>T</mi><mn>0</mn></msub><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>p</mi><mo>=</mo><msub><mi>ρ</mi><mi>B</mi></msub><mfenced><mfrac><mi>R</mi><mi>M</mi></mfrac></mfenced><mn>3</mn><msub><mi>T</mi><mn>0</mn></msub><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>iii</mi></mfenced></math></p><p>From Equations. (ii) and (iii), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mfrac><msub><mi>ρ</mi><mi>A</mi></msub><msub><mi>ρ</mi><mi>B</mi></msub></mfrac><mo>·</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>ρ</mi><mi>B</mi></msub><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msub><mi>ρ</mi><mi>A</mi></msub><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msub><mi>ρ</mi><mn>0</mn></msub></math></p></div>


tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
title:
the-probability-density-plots-of-1-s-and-2-s-orbitals-are-given-in-figure-the-density-of-dots-in-a-region-represents-the-probability-density-of-findin-1

slug:
the-probability-density-plots-of-1-s-and-2-s-orbitals-are-given-in-figure-the-density-of-dots-in-a-region-represents-the-probability-density-of-findin-1-86068

question:
<div class="question">The probability density plots of $1 s$ and $2 s$ orbitals are given in figure.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/KW6UlqtmYx8qU0E2Ze00YtyIq-_2FHH_3i0-1RRE1OQ.original.fullsize.png"/><br/><br/>The density of dots in a region represents the probability density of finding electrons in the region.<br/>On the basis of above diagram which of the following statements is incorrect?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 s$ and $2 s$ orbitals are spherical in shape.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The probability of finding the electron is maximum near the nucleus.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The probability of finding the electron at a given distance is equal in all directions.</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">The probability density of electrons for $2 s$ orbital decreases uniformly as distance from the nucleus increases.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The probability density of electrons for $2 s$ orbital decreases uniformly as distance from the nucleus increases.</span> </div>

answer:
<div class="solution">The probability density of electrons in $2 s$ orbital first increases then decreases and after that it increases again as distance increases from nucleus.</div>


title: the-probability-of-getting-sum-more-than-7-when-a-pair-of-dice-are-thrown-is
slug: the-probability-of-getting-sum-more-than-7-when-a-pair-of-dice-are-thrown-is-32236
tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The probability of getting sum more than 7 when a pair of dice are thrown is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{7}{36}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{5}{12}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{7}{12}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{12}$</span> </div>

answer:
<div class="solution">Here $n(S)=6^2=36$<br/>Let $\mathrm{E}$ be the event "getting sum more than 7" i.e. sum of pair of dice $=8,9,10,11,12$<br/>i.e. $E=\left\{\begin{array}{lllll}(2,6) &amp; (3,5) &amp; (4,4) &amp; (5,3) &amp; (6,2) \\ (3,6) &amp; (4,5) &amp; (5,4) &amp; (6,3) \\ (4,6) &amp; (5,5) &amp; (6,4) &amp; \\ (5,6) &amp; (6,5) &amp; (6,6) &amp; \end{array}\right\}$<br/>$\therefore \quad n(E)=15 . \therefore$ Req. prob $=\frac{n(E)}{n(S)}=\frac{15}{36}=\frac{5}{12}$</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
title:
the-probability-of-india-winning-a-test-match-against-west-indies-is-12-assuming-independence-from-match-to-match-the-probability-that-in-a-5-match-se

slug:
the-probability-of-india-winning-a-test-match-against-west-indies-is-12-assuming-independence-from-match-to-match-the-probability-that-in-a-5-match-se-69041

question:
<div class="question">The probability of India winning a test match against West Indies is $1 / 2$. Assuming independence from match to match, the probability that in a 5 match series India's second win occurs at the third test, is -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 / 3$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 / 2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1 / 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 / 8$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 4$</span> </div>

answer:
<div class="solution">The sample space is [LWW, WLW]
<br/> <br/>$$
<br/> <br/>\therefore P(\mathrm{LWW})+\mathrm{P}(\mathrm{WLW})
<br/> <br/>$$
<br/> <br/>= Probability that in 5 match series, it is India's second win
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>=P(\mathrm{~L}) \mathrm{P}(\mathrm{W}) \mathrm{P}(\mathrm{W})+\mathrm{P}(\mathrm{W}) \mathrm{P}(\mathrm{L}) \mathrm{P}(\mathrm{W}) \\
<br/> <br/>=\frac{1}{8}+\frac{1}{8}=\frac{2}{8}=\frac{1}{4}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
title:
the-probability-of-simultaneous-occurrence-of-atleast-one-of-two-events-a-and-b-is-p-if-the-probability-that-exactly-one-of-a-b-occurs-is-q-then-p-a-p

slug:
the-probability-of-simultaneous-occurrence-of-atleast-one-of-two-events-a-and-b-is-p-if-the-probability-that-exactly-one-of-a-b-occurs-is-q-then-p-a-p-92748

question:
<div class="question">The probability of simultaneous occurrence of atleast one of two events $A$ and $B$ is $p .$ If the probability that exactly one of $A, B$ occurs is $q$ then $P\left(A^{\prime}\right)+P\left(B^{\prime}\right)$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2-2 p+q$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2+2 p-q$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3-3 p+q$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2-p+q$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2-2 p+q$</span> </div>

answer:
<div class="solution">Since, $P($ exactly one of $A, B$ occurs $)=q$.
<br/>$\therefore \quad P(A \cup B)-P(A \cap B)=q$
<br/>$\Rightarrow \quad p-P(A \cap B)=q \Rightarrow P(A \cap B)=p-q$
<br/>$\Rightarrow \quad 1-P\left(A^{\prime} \cup B^{\prime}\right)=p-q \Rightarrow P\left(A^{\prime} \cup B^{\prime}\right)=1-p+q$
<br/>$\Rightarrow \quad P\left(A^{\prime}\right)+P\left(B^{\prime}\right)-P\left(A^{\prime} \cap B^{\prime}\right)=1-p+q$
<br/>$\Rightarrow \quad P\left(A^{\prime}\right)+P\left(B^{\prime}\right)=(1-p+q)+[1-P(A \cup B)]$
<br/>$=(1-p+q)+(1-p)=2-2 p+q$</div>


title: the-probability-that-a-card-drawn-from-a-pack-of-52-cards-will-be-a-diamond-or-king-is
slug: the-probability-that-a-card-drawn-from-a-pack-of-52-cards-will-be-a-diamond-or-king-is-87335
tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The probability that a card drawn from a pack of 52 cards will be a diamond or king is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{52}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2}{13}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{4}{13}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{13}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4}{13}$</span> </div>

answer:
<div class="solution">Total no. of cards $=52$<br/>13 cards are diamonds and 4 cards are king.<br/>There is only one card which is a king of diamond.<br/>$\therefore \mathrm{P}($ card is diamond $)=\frac{13}{52}, \mathrm{P}$ (card is king)<br/>$=\frac{4}{52}$<br/>$\mathrm{P}($ card is king of diamond $)=\frac{1}{52}$<br/>$\therefore \quad P$ (card is diamond or king)<br/>$=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{16}{52}=\frac{4}{13}$</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.891</span> </div>
source_db: MarksBatch2_P3.db
title:
the-probability-that-certain-electronic-component-fails-when-first-used-is-010-if-it-does-not-fail-immediately-the-probability-that-it-lasts-for-one-y

slug:
the-probability-that-certain-electronic-component-fails-when-first-used-is-010-if-it-does-not-fail-immediately-the-probability-that-it-lasts-for-one-y-34028

question:
<div class="question">The probability that certain electronic component fails when first used is 0.10 . If it does not fail immediately, the probability that it lasts for one year is $0.99 .$ The probability that a new component will last for one year is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.99</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.871</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0.891</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.762</span> </li> </ul>

answer:
<div class="solution">Probability that the electronic component fails when first used is $P(F)=0.10$. Therefore, $P\left(F^{\prime}\right)=1-P(F)=0.90$
<br/> <br/>Let $E$ be the event that a new component will last for one year.
<br/> <br/>$\text { Then, } P(E)=P(F) P\left(\frac{E}{F}\right)+P\left(F^{\prime}\right) P\left(\frac{E}{F^{\prime}}\right)$
<br/> <br/>[total probability theorem] $=0.10 \times 0+0.90 \times 0.99=0.891$</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
title:
the-product-of-atomic-weight-and-specific-heat-of-any-element-is-a-constant-approximately-64-this-is-known-as

slug:
the-product-of-atomic-weight-and-specific-heat-of-any-element-is-a-constant-approximately-64-this-is-known-as-20398

question:
<div class="question">The product of atomic weight and specific heat of any element is a constant, approximately $6.4 .$ This is known as:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Dalton's law</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Avogadro's law</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Newton's law</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Dulong Pettit law</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Dulong Pettit law</span> </div>

answer:
<div class="solution">According to Dulong and Pettit's law<br/>Atomic weight $\times$ Specific heat $=6.4$ (approx) This law is applicable only to solid elements but it fails to explain very high specific heat of diamond.</div>


title: the-product-of-n-positive-numbers-is-unity-then-their-sum-is
slug: the-product-of-n-positive-numbers-is-unity-then-their-sum-is-63260
question: <div class="question">The product of $n$ positive numbers is unity, then their sum is:</div>
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a positive integer</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">divisible by $n$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">equal to $n+\frac{1}{n}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">never less than $n$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">never less than $n$</span> </div>

answer:
<div class="solution">Since, product of $n$ positive number is unity. $\Rightarrow x_{1} x_{2} x_{3} \ldots \ldots x_{n}=1$
<br/> <br/>Using A.M. $\geq$ GM
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow \frac{x_{1}+x_{2}+\ldots . .+x_{n}}{n} \geq\left(x_{1} x_{2} \ldots \ldots x_{n}\right)^{\frac{1}{n}} \\
<br/> <br/>\Rightarrow x_{1}+x_{n}+\ldots \ldots+x_{n} \geq n(1)^{\frac{1}{n}}\left[\text { From eq }^{\mathrm{n}}(\mathrm{i})\right]
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
title:
the-quadratic-equation-whose-roots-are-the-x-and-yintercepts-of-the-line-passing-through-11-and-making-a-triangle-of-area-a-with-the-coordinate-axes-i

slug:
the-quadratic-equation-whose-roots-are-the-x-and-yintercepts-of-the-line-passing-through-11-and-making-a-triangle-of-area-a-with-the-coordinate-axes-i-50242

question:
<div class="question">The quadratic equation whose roots are the $\mathrm{x}$ and yintercepts of the line passing through (1,1) and making a triangle of area $\mathrm{A}$ with the co-ordinate axes is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+A x+2 A=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x^{2}-2 A x+2 A=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x^{2}-A x+2 A=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2}-2 A x+2 A=0$</span> </div>

answer:
<div class="solution">Equation of the line making intercepts a and
<br/> <br/>$b$ on the axes is $\frac{x}{a}+\frac{y}{b}=1$ Since, it passes through (1,1) $\Rightarrow \frac{1}{\mathrm{a}}+\frac{1}{\mathrm{~b}}=1$
<br/> <br/>Also the area of the triangle formed by the line and the axes is $\mathrm{A}$.
<br/> <br/>$\therefore \frac{1}{2} \mathrm{ab}=\mathrm{A} \Rightarrow \mathrm{ab}=2 \mathrm{~A}$
<br/> <br/>From eqs. (i) and (ii), we get, $a+b=2 A$ Hence, a and b are the roots of the eq. $x^{2}-(a+b) x+a b=0 \Rightarrow x^{2}-2 A x+2 A=0$</div>


title: the-radiation-emitted-by-a-perfectly-black-body-is-proportional-to
slug: the-radiation-emitted-by-a-perfectly-black-body-is-proportional-to-17779
question: <div class="question">The radiation emitted by a perfectly black body is proportional to</div>
tags: ['Physics', 'Thermal Properties of Matter', 'BITSAT', 'BITSAT 2011']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">temperature on ideal gas scale</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">fourth root of temperature on ideal gas scale</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">fourth power of temperature on ideal gas scale</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">square of temperature on ideal gas scale</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">fourth power of temperature on ideal gas scale</span> </div>


tags: ['Mathematics', 'Properties of Triangles', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P3.db
title:
the-random-variable-mathrmx-has-the-following-probability-distribution-beginarraycccccc-hline-mathrmx-0-1-2-3-4-hline-mathrmpmathrmxmathrmx-mathrmk-3-

slug:
the-random-variable-mathrmx-has-the-following-probability-distribution-beginarraycccccc-hline-mathrmx-0-1-2-3-4-hline-mathrmpmathrmxmathrmx-mathrmk-3-82507

question:
<div class="question">The random variable \(\mathrm{X}\) has the following probability distribution<br/>\(\begin{array}{|c|c|c|c|c|c|}<br/>\hline \mathrm{x} &amp; 0 &amp; 1 &amp; 2 &amp; 3 &amp; 4 \\<br/>\hline \mathrm{P}(\mathrm{X}=\mathrm{x}) &amp; \mathrm{k} &amp; 3 \mathrm{k} &amp; 5 \mathrm{k} &amp; 2 \mathrm{k} &amp; \mathrm{k} \\<br/>\hline<br/>\end{array}\)<br/>Then the value of \(P(X \geq 2)\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{1}{3}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{2}{3}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{3}{4}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{1}{4}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{2}{3}\)</span> </div>

answer:
<div class="solution">Since, \(\Sigma \mathrm{P}_{\mathrm{i}}(\mathrm{X}=\mathrm{x})=1\)<br/>\(\begin{aligned}<br/>&amp; \therefore \mathrm{k}+3 \mathrm{k}+5 \mathrm{k}+2 \mathrm{k}+\mathrm{k}=1 \therefore 12 \mathrm{k}=1 \therefore \mathrm{k}=\frac{1}{12} \\<br/>&amp; \text { Now, } \mathrm{P}(\mathrm{X} \geq 2)=\mathrm{P}(\mathrm{X}=2)+\mathrm{P}(\mathrm{X}=3)+\mathrm{P}(\mathrm{X}=4) \\<br/>&amp; =5 \mathrm{k}+2 \mathrm{k}+\mathrm{k}=8 \mathrm{k}=8\left(\frac{1}{12}\right)=\frac{2}{3}<br/>\end{aligned}\)</div>


title: the-range-of-the-function-f-x-3-x-2-4-x-5-is
slug: the-range-of-the-function-f-x-3-x-2-4-x-5-is-35227
question: <div class="question">The range of the function $f(x)=\sqrt{3 x^2-4 x+5}$ is</div>
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> $\left(-\infty, \sqrt{\frac{11}{3}}\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(-\infty, \sqrt{\frac{11}{5}}\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left[\sqrt{\frac{11}{3}}, \infty\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\sqrt{\frac{11}{5}}, \infty\right)$.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\sqrt{\frac{11}{3}}, \infty\right)$</span> </div>

answer:
<div class="solution">$f(x)$ is defined if $3 x^2-4 x+5 \geq 0$<br/>$\Rightarrow 3\left[x^2-\frac{4}{3} x+\frac{5}{3}\right] \geq 0 \Rightarrow 3\left[\left(x-\frac{2}{3}\right)^2+\frac{11}{9}\right] \geq 0$<br/>Which is true for all real $x \therefore$ Domain of $f(x)=$ $(-\infty, \infty)$<br/>Let $y=\sqrt{3 x^2-4 x+5}$<br/>$\Rightarrow y^2=3 x^2-4 x+5$ i.e. $3 x^2-4 x+\left(5-y^2\right)=0$<br/>For $x$ to be real , $16-12\left(5-y^2\right) \geq 0 \Rightarrow y \geq \sqrt{\frac{11}{3}}$<br/>$\therefore \quad$ Range of $y=\left[\sqrt{\frac{11}{3}}, \infty\right)$</div>


title: the-range-of-the-function-fxfracx2x1x2x1-where-mathrmx-in-mathrmr-is
slug: the-range-of-the-function-fxfracx2x1x2x1-where-mathrmx-in-mathrmr-is-35369
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P3.db
question:
<div class="question">The range of the function \(f(x)=\frac{x^2-x+1}{x^2+x+1}\) where \(\mathrm{x} \in \mathrm{R}\), is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((-\infty, 3]\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\((-\infty, \infty)\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\([3, \infty)\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(\left[\frac{1}{3}, 3\right]\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\left[\frac{1}{3}, 3\right]\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Let } y=\frac{x^2-x+1}{x^2+x+1} \\<br/>&amp; \Rightarrow \mathrm{x}^2(\mathrm{y}-1)+\mathrm{x}(\mathrm{y}+1)+(\mathrm{y}-1)=0 \\<br/>&amp; \Rightarrow x=\frac{-(y+1) \pm \sqrt{(y+1)^2-4(y-1)^2}}{2(y-1)} \\<br/>&amp; =\frac{-(y+1) \pm \sqrt{-3 y^2+10 y-3}}{2(y-1)} \text { is real iff } \\<br/>&amp; \mathrm{y}-1 \neq 0 \Rightarrow \mathrm{y} \neq 1 \\<br/>&amp; \text { If } y=1 \text { then original equation gives } \mathrm{x}=0 \text {, } \\<br/>&amp; \text { so taking } \mathrm{y}=1 \\<br/>&amp; \text { Also } 3 y^2-10 y+3 \leq 0 \\<br/>&amp; \Rightarrow(3 y-1)(y-3) \leq 0 \\<br/>&amp; \Rightarrow \mathrm{y} \in\left[\frac{1}{3}, 3\right] \quad \therefore \text { Range is }\left[\frac{1}{3}, 3\right] \\<br/>&amp;<br/>\end{aligned}\)</div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
title:
the-range-of-the-particle-when-launched-at-an-angle-of-1-5-with-the-horizontal-is-15-km-what-is-the-range-of-the-projectile-when-launched-at-an-angle-

slug:
the-range-of-the-particle-when-launched-at-an-angle-of-1-5-with-the-horizontal-is-15-km-what-is-the-range-of-the-projectile-when-launched-at-an-angle-16194

question:
<div class="question">The range of the particle when launched at an angle of $15^{\circ}$ with the horizontal is $1.5 \mathrm{~km}$. What is the range of the projectile when launched at an angle of $45^{\circ}$ to the horizontal.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.5 \mathrm{~km}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$3.0 \mathrm{~km}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$6.0 \mathrm{~km}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.75 \mathrm{~km}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3.0 \mathrm{~km}$</span> </div>

answer:
<div class="solution">$1.5=\frac{\mathrm{u}^{2} \sin 30}{\mathrm{~g}} ; \mathrm{R}=\frac{\mathrm{u}^{2} \sin 90}{\mathrm{~g}}=3 \mathrm{~km}$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
title:
the-rate-of-a-reaction-triples-when-temperature-changes-from-20-0-c-to-50-0-c-the-energy-of-activation-for-the-reaction-is-r-8-314-jk-1-mol-1

slug:
the-rate-of-a-reaction-triples-when-temperature-changes-from-20-0-c-to-50-0-c-the-energy-of-activation-for-the-reaction-is-r-8-314-jk-1-mol-1-32255

question:
<div class="question">The rate of a reaction triples when temperature changes from <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>20</mn><mn>0</mn></msup><mi mathvariant="normal">C</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>50</mn><mn>0</mn></msup><mi mathvariant="normal">C</mi></math>. The energy of activation for the reaction is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="normal">R</mi><mo>=</mo><mn>8</mn><mo>.</mo><mn>314</mn><msup><mi>JK</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mi>mol</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>181</mn><mo>.</mo><mn>327</mn><mo> </mo><mi mathvariant="normal">J</mi><mo> </mo><msup><mi>mol</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>428</mn><mo>.</mo><mn>141</mn><mo> </mo><mi mathvariant="normal">J</mi><mo> </mo><msup><mi>mol</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn><mo>.</mo><mn>4321</mn><msup><mi>kJmol</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn><mo>.</mo><mn>8118</mn><msup><mi>kJmol</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn><mo>.</mo><mn>8118</mn><msup><mi>kJmol</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></span> </div>

answer:
<div class="solution"><p>Arrhenius equation is given by,<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mfrac><msub><mi mathvariant="normal">K</mi><mn>2</mn></msub><msub><mi mathvariant="normal">K</mi><mn>1</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi mathvariant="normal">E</mi><mi mathvariant="normal">a</mi></msub><mrow><mn>2</mn><mo>.</mo><mn>303</mn><mo>×</mo><mi mathvariant="normal">R</mi></mrow></mfrac><mfenced close="]" open="["><mfrac><mrow><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub><mo>-</mo><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub></mrow><mrow><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub></mrow></mfrac></mfenced></math></p><p>Given,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi mathvariant="normal">K</mi><mn>2</mn></msub><msub><mi mathvariant="normal">K</mi><mn>1</mn></msub></mfrac><mo>=</mo><mn>3</mn><mo>;</mo><mi mathvariant="normal">R</mi><mo>=</mo><mo>-</mo><mn>8</mn><mo>.</mo><mn>314</mn><mo> </mo><msup><mi>JK</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>mol</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>1</mn></msub><mo>=</mo><mn>20</mn><mo>+</mo><mn>273</mn><mo>=</mo><mn>293</mn><mo> </mo><mi mathvariant="normal">K</mi></math><br/>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mn>2</mn></msub><mo>=</mo><mn>50</mn><mo>+</mo><mn>273</mn><mo>=</mo><mn>323</mn><mo> </mo><mi mathvariant="normal">K</mi></math></p><p>Substituting the given values in Arrhenius equation,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mn>3</mn><mo>=</mo><mfrac><msub><mi mathvariant="normal">E</mi><mi mathvariant="normal">a</mi></msub><mrow><mn>8</mn><mo>.</mo><mn>314</mn><mo>×</mo><mn>2</mn><mo>.</mo><mn>303</mn></mrow></mfrac><mfenced close="]" open="["><mfrac><mrow><mn>323</mn><mo>-</mo><mn>293</mn></mrow><mrow><mn>323</mn><mo>×</mo><mn>293</mn></mrow></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">E</mi><mi mathvariant="normal">a</mi></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mo>.</mo><mn>303</mn><mo>×</mo><mn>8</mn><mo>.</mo><mn>314</mn><mo>×</mo><mn>323</mn><mo>×</mo><mn>293</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>477</mn></mrow><mn>30</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>28811</mn><mo>.</mo><mn>8</mn><mo> </mo><mi mathvariant="normal">J</mi><mo> </mo><msup><mi>mol</mi><mn>1</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>28</mn><mo>.</mo><mn>8118</mn><msup><mi>kJmol</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p></div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P3.db
title:
the-ratio-in-which-the-yzplane-divide-the-line-segment-formed-by-joining-the-points-2-4-7-and-3-5-8-is-2-m-the-value-of-m-is

slug:
the-ratio-in-which-the-yzplane-divide-the-line-segment-formed-by-joining-the-points-2-4-7-and-3-5-8-is-2-m-the-value-of-m-is-36851

question:
<div class="question">The ratio in which the YZ-plane divide the line segment formed by joining the points $(-2,4,7)$ and $(3,-5,8)$ is $2: \mathrm{m}$. The value of $\mathrm{m}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">Let the points be $A(-2,4,7)$ and $B(3,-5,8)$ on $Y Z$-plane, $x$-coordinate $=0$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/U3gdRe4rjERxY6nARZ6_prnRjGEwHH0AQrCCMdLDNDM.original.fullsize.png"/><br/><br/>Let the ratio be $K: 1$. <br/>The coordinates of $C$ are $\left(\frac{3 K-2}{K+1}, \frac{-5 K+4}{K+1}, \frac{8 K+7}{K+1}\right)$<br/>Clearly $\frac{3 K-2}{K+1}=0 \Rightarrow 3 K=2 \Rightarrow K=\frac{2}{3}$ <br/>Hence required ratio is $2: 3$.</div>


title: the-ratio-k-c-k-p-for-the-reaction-co-g-2-1-o-2-g-co-2-g-is
slug: the-ratio-k-c-k-p-for-the-reaction-co-g-2-1-o-2-g-co-2-g-is-77901
tags: ['Chemistry', 'Chemical Equilibrium', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
question:
<div class="question">The ratio $\frac{K_{p}}{K_{c}}$ for the reaction
<br/> <br/>$\mathrm{CO}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \rightleftharpoons \mathrm{CO}_{2}(g)$ is:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{\sqrt{R T}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(R T)^{1 / 2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{RT}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\sqrt{R T}}$</span> </div>

answer:
<div class="solution">$K_{P}=K_{C}(R T)^{\Delta n_{\text {g }}}$
<br/> <br/>For the reaction
<br/> <br/>$\mathrm{CO}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(g) \leftrightharpoons \mathrm{CO}_{2}(\mathrm{~g})$
<br/> <br/>$\Delta n_{g}=1-\left(1+\frac{1}{2}\right)=-\frac{1}{2}$
<br/> <br/>$\therefore \quad \mathrm{K}_{\mathrm{P}}=\frac{\mathrm{K}_{\mathrm{C}}}{\sqrt{\mathrm{RT}}} ; \quad \frac{\mathrm{K}_{\mathrm{P}}}{\mathrm{K}_{\mathrm{C}}}=\frac{1}{\sqrt{\mathrm{RT}}}$</div>


title: the-ratio-of-angular-momentum-l-to-the-atomic-dipole-moment-i-for-hydrogen-like-atoms-and-ions-is
tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
slug:
the-ratio-of-angular-momentum-l-to-the-atomic-dipole-moment-i-for-hydrogen-like-atoms-and-ions-is-52521

question:
<div class="question">The ratio of angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> to the atomic dipole moment <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>i</mi></msub></math> for hydrogen like atoms and ions is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">always constant and is equal to the ratio of mass to the charge of electron</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">always constant and is equal to twice the ratio of mass to the charge of electron to the principal quantum number <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">proportional to the principal quantum number <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">proportional to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>n</mi><mn>2</mn></msup></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">always constant and is equal to the ratio of mass to the charge of electron</span> </div>

answer:
<div class="solution"><p>The electron moving with a speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> in the circular Bohr orbit of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> constitutes a current of magnitude.</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/a7b5a0a6-b400-4706-8f72-baaf9b8de79c-image.png" style="width: 200px; height: 232px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mfrac><mi>e</mi><mi>T</mi></mfrac><mo>.</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is the orbital period of the electron.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>2</mn><mi>π</mi><mi>r</mi><mo>/</mo><mi>v</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mfrac><mrow><mi>e</mi><mi>v</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>r</mi></mrow></mfrac></math></p><p>By definition of atomic dipole moment <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>i</mi></msub><mo>,</mo></math> its magnitude is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>i</mi></msub><mo>=</mo><mi>i</mi><mi>A</mi><mo>=</mo><mfrac><mrow><mi>e</mi><mi>V</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>r</mi></mrow></mfrac><mo>·</mo><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mi>e</mi><mi>v</mi><mi>r</mi></mrow><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Angular momentum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mi>m</mi><mi>v</mi><mi>r</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>L</mi><msub><mi>μ</mi><mi>i</mi></msub></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>m</mi><mi>v</mi><mi>r</mi></mrow><mrow><mi>e</mi><mi>v</mi><mi>r</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>m</mi></mrow><mi>e</mi></mfrac><mo>=</mo><mtext> constant </mtext></math></p></div>


tags: ['Physics', 'Electromagnetic Waves', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
title:
the-ratio-of-average-electric-energy-density-and-total-average-energy-density-of-electromagnetic-wave-is-1

slug:
the-ratio-of-average-electric-energy-density-and-total-average-energy-density-of-electromagnetic-wave-is-1-78980

question:
<div class="question">The ratio of average electric energy density and total average energy density of electromagnetic wave is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">We have $\frac{\mathrm{U}_{\mathrm{E}}}{\mathrm{U}_{\mathrm{T}}}=\frac{\mathrm{U}_{\mathrm{E}}}{\mathrm{U}_{\mathrm{E}}+\mathrm{U}_{\mathrm{B}}}=\frac{\mathrm{U}_{\mathrm{E}}}{2 \mathrm{U}_{\mathrm{E}}}=\frac{1}{2}$<br/>$\left[\because \mathrm{U}_{\mathrm{E}}=\mathrm{U}_{\mathrm{B}}=\frac{1}{2} \in_0 \mathrm{E}_0^2=\frac{\mathrm{B}_0^2}{2 \mu_0}\right]$</div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$3: 2$</span> </div>
source_db: MarksBatch2_P3.db
title:
the-ratio-of-intensities-at-two-points-p-and-q-on-the-screen-in-a-youngs-double-slit-experiment-where-phase-difference-between-two-wave-of-same-amplit

slug:
the-ratio-of-intensities-at-two-points-p-and-q-on-the-screen-in-a-youngs-double-slit-experiment-where-phase-difference-between-two-wave-of-same-amplit-62308

question:
<div class="question">The ratio of intensities at two points $\mathrm{P}$ and $\mathrm{Q}$ on the screen in a Young's double slit experiment where phase difference between two wave of same amplitude are $\frac{\pi}{3}$ and $\frac{\pi}{2}$, respectively are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1: 3$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3: 1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$3: 2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2: 3$</span> </li> </ul>

answer:
<div class="solution">Intensity at a point in Young's double slit experiment is given by<br/>$\mathrm{I}=\mathrm{I}_1+\mathrm{I}_2+2 \sqrt{\mathrm{I}_1 \mathrm{I}_2} \cos \phi$<br/>Here $\mathrm{I}_1=\mathrm{I}_2=\mathrm{I}_0$ (say)<br/>At $P$<br/>$\therefore \mathrm{I}_{\mathrm{p}}=\mathrm{I}_0+\mathrm{I}_0+2 \mathrm{I}_0 \cos \frac{\pi}{3}=2 \mathrm{I}_0+2 \mathrm{I}_0 \times \frac{1}{2}=3 \mathrm{I}_0$<br/>At $Q$<br/>$\mathrm{I}_{\mathrm{Q}}=\mathrm{I}_0+\mathrm{I}_0+2 \mathrm{I}_0 \cos 90^{\circ}=2 \mathrm{I}_0$<br/>$\frac{\mathrm{I}_{\mathrm{P}}}{\mathrm{I}_{\mathrm{Q}}}=\frac{3}{2}$</div>


tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
title:
the-ratio-of-intensity-at-the-centre-of-a-bright-fringe-to-the-intensity-at-a-point-distance-onefourth-of-the-distance-between-two-successive-bright-f

slug:
the-ratio-of-intensity-at-the-centre-of-a-bright-fringe-to-the-intensity-at-a-point-distance-onefourth-of-the-distance-between-two-successive-bright-f-27859

question:
<div class="question">The ratio of intensity at the centre of a bright fringe to the intensity at a point distance one-fourth of the distance between two successive bright fringes will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li> </ul>

answer:
<div class="solution"><p>Intensity at centre of bright fringe,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mn>0</mn></msub><mo>=</mo><mi>I</mi><mo>+</mo><mi>I</mi><mo>+</mo><mn>2</mn><msqrt><mi>I</mi><mi>I</mi></msqrt><mo>·</mo><mi>cos</mi><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mn>0</mn></msub><mo>=</mo><mn>2</mn><mi>I</mi><mo>+</mo><mn>2</mn><mi>I</mi></math>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>∵</mo><mo> </mo><mi>cos</mi><mn>0</mn><mo>°</mo><mo>=</mo><mn>1</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mi>I</mi></math></p><p>Similarly Intensity at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>P</mi><mn>4</mn></mfrac></math></p><p>(with phase difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mn>4</mn></mfrac><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math>)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>I</mi><mo>'</mo></msup><mo>=</mo><mi>I</mi><mo>+</mo><mi>I</mi><mo>+</mo><mn>2</mn><msqrt><mi>I</mi><mi>I</mi></msqrt><mi>cos</mi><mfrac><mi>π</mi><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mi>I</mi><mo>+</mo><mn>2</mn><msqrt><mi>I</mi><mi>I</mi></msqrt><mo>×</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mi>I</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><msub><mi>I</mi><mn>0</mn></msub><msup><mi>I</mi><mo>'</mo></msup></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><mi>I</mi></mrow><mrow><mn>2</mn><mi>I</mi></mrow></mfrac><mo>=</mo><mn>2</mn></math></p></div>


title: the-ratio-of-oxidation-states-of-cl-in-potassium-chloride-to-that-in-potassium-chlorate-is
slug: the-ratio-of-oxidation-states-of-cl-in-potassium-chloride-to-that-in-potassium-chlorate-is-23904
tags: ['Chemistry', 'Redox Reactions', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
question:
<div class="question">The ratio of oxidation states of $\mathrm{Cl}$ in potassium chloride to that in potassium chlorate is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{+1}{5}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{-1}{5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{-2}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{+3}{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{-1}{5}$</span> </div>

answer:
<div class="solution">\mathrm{KCl} x &amp; =-1 &amp; \mathrm{KClO}_{3}
<br/> <br/>potassium chloride $+1+x-6=0$
<br/> <br/>$$
<br/> <br/>x=+5
<br/> <br/>$$
<br/> <br/>potassium chlorate.
<br/> <br/>$\therefore$ Ratio of oxidation state of $\mathrm{Cl}=\frac{-1}{5}$</div>


tags: ['Physics', 'Electromagnetic Induction', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
title:
the-ratio-of-secondary-and-primary-turns-of-stepup-transformer-is-4-1-if-a-current-of-4-a-is-applied-to-the-primary-the-induced-current-in-secondary-w

slug:
the-ratio-of-secondary-and-primary-turns-of-stepup-transformer-is-4-1-if-a-current-of-4-a-is-applied-to-the-primary-the-induced-current-in-secondary-w-44966

question:
<div class="question">The ratio of secondary and primary turns of step-up transformer is $4: 1 .$ If a current of 4 A is applied to the primary, the induced current in secondary will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$8 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.5 \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 \mathrm{~A}$</span> </div>

answer:
<div class="solution">$\frac{\mathrm{I}_{\mathrm{S}}}{\mathrm{I}_{\mathrm{P}}}=\frac{\mathrm{N}_{\mathrm{P}}}{\mathrm{N}_{\mathrm{S}}}=\frac{1}{4} \Rightarrow \mathrm{I}_{\mathrm{S}}=\frac{1}{4} \times 4=1 \mathrm{~A}$</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$4: 1$</span> </div>
source_db: MarksBatch2_P3.db
title:
the-ratio-of-speed-of-sound-in-hydrogen-gas-to-the-speed-of-sound-in-oxygen-gas-at-the-same-temperature-is-1

slug:
the-ratio-of-speed-of-sound-in-hydrogen-gas-to-the-speed-of-sound-in-oxygen-gas-at-the-same-temperature-is-1-22148

question:
<div class="question">The ratio of speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$4: 1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1: 2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1: 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1: 1$</span> </li> </ul>

answer:
<div class="solution">Given $\mathrm{M}_{\mathrm{H}_2}=2 ; \mathrm{M}_{\mathrm{O}_2}=32$<br/>Speed of sound, $\mathrm{v}=\sqrt{\frac{\gamma \mathrm{RT}}{\mathrm{M}}}$<br/>$\Rightarrow \quad v \propto \frac{1}{\sqrt{M}}$<br/>$\therefore \quad \frac{\mathrm{v}_{\mathrm{H}_2}}{\mathrm{v}_{\mathrm{O}_2}}=\sqrt{\frac{\mathrm{M}_{\mathrm{O}_2}}{\mathrm{M}_{\mathrm{H}_2}}}=\sqrt{\frac{32}{2}}=4: 1$</div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5: 7</span> </div>
source_db: MarksBatch2_P3.db
title:
the-ratio-of-the-accelerations-for-a-solid-sphere-mass-m-and-radius-r-rolling-down-an-incline-of-angle-without-slipping-and-slipping-down-the-incline-

slug:
the-ratio-of-the-accelerations-for-a-solid-sphere-mass-m-and-radius-r-rolling-down-an-incline-of-angle-without-slipping-and-slipping-down-the-incline-25017

question:
<div class="question">The ratio of the accelerations for a solid sphere (mass 'm' and radius 'R') rolling down an incline of angle ' $\theta$ ' without slipping and slipping down the incline without rolling is:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">5: 7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2: 3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2: 5</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">7: 5</span> </li> </ul>

answer:
<div class="solution">For solid sphere rolling without slipping on inclined plane, acceleration<br/>$$<br/>a_{1}=\frac{g \sin \theta}{1+\frac{K^{2}}{R^{2}}}<br/>$$<br/>For solid sphere slipping on inclined plane without rolling, acceleration<br/>$$<br/>a_{2}=g \sin \theta<br/>$$<br/>Therefore required ratio $=\frac{\mathrm{a}_{1}}{\mathrm{a}_{2}}$<br/>$$<br/>=\frac{1}{1+\frac{K^{2}}{R^{2}}}=\frac{1}{1+\frac{2}{5}}=\frac{5}{7}<br/>$$</div>


title: the-reaction-2
slug: the-reaction-2-14776
tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question">The reaction<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/hfzCedRV0rr6N7cEREtAN7KUTz89Z8TQzKmop22MZ8g.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Rosenmund's reaction</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>Stephen's reaction</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>Cannizzaro's reaction</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>Gatterman-Koch reaction</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Gatterman-Koch reaction</span> </div>

answer:
<div class="solution">This reaction proceeds in the presence of anhydrous $\mathrm{AlCl}_3$ or $\mathrm{CuCl}$.</div>


title: the-reaction-is-called-2
slug: the-reaction-is-called-2-74249
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
question:
<div class="question">The reaction<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/MmBRaTJE8GZ9dpjjwpWr3besScClqZXWhkhXf8tJUAs.original.fullsize.png"/><br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ykMhi1N51MZc66PK6S0OpvK7rtnxsf6EffSuf7RTShk.original.fullsize.png"/><br/><br/>is called:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Williamson continuous etherification process</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Etard reaction</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Gatterman - Koch reaction</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Williamson Synthesis</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Williamson Synthesis</span> </div>

answer:
<div class="solution">Williamson synthesis is one of the best methods for the preparation of symmetrical and unsymmetrical ethers. In this method, an alkyl halide is allowed to react with sodium alkoxide.</div>


title: the-reaction-is-known-as-1
slug: the-reaction-is-known-as-1-86624
tags: ['Chemistry', 'Carboxylic Acid Derivatives', 'BITSAT', 'BITSAT 2014']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The reaction,
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/fsk_9i1672XfeKaR3FUlrXnuMQjhSp6U9dkunNObnKI.original.fullsize.png"/><br/> <br/> <br/>is known as:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Wurtz reaction</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Koch reaction</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Clemmensen reduction</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Kolbe's reaction</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Koch reaction</span> </div>


title: the-reaction-of-c-6-h-5-ch-chch-3-with-hbr-produces-1
slug: the-reaction-of-c-6-h-5-ch-chch-3-with-hbr-produces-1-48058
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question">The reaction of $\mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}=\mathrm{CHCH}_3$ with $\mathrm{HBr}$ produces</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/sM8g3R2Lowv8hh07ko9elgOQu8s2lyEEt60k0gXDo60.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/d2t-MC2jj-ALM5nfME5ftFVvnn28HrUdGnUm8HNvJWM.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{CH}_2 \mathrm{Br}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/mu9NY6eP_5X7LZJgV6EEj0Axi2Tgc5lL7pStXK1ERqg.original.fullsize.png"/><br/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/sM8g3R2Lowv8hh07ko9elgOQu8s2lyEEt60k0gXDo60.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/QtPBDHH146Bbfl6e2FuBdjGhqPN9E0mPdeTjU1smme8.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/wHeqdZihl5o9_Ax_v4OvyRkdpwtULxXoqTnVWLhrzqw.original.fullsize.png"/><br/> <br/>Electrophile addition reaction takes place via more stable carbocation.</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
title:
the-reaction-so-2-cl-2-so-2-cl-2-is-exothermic-and-reversible-a-mixture-of-so-2-g-cl-2-g-and-so-2-cl-2-l-is-at-equilibrium-in-a-closed-container-now-a

slug:
the-reaction-so-2-cl-2-so-2-cl-2-is-exothermic-and-reversible-a-mixture-of-so-2-g-cl-2-g-and-so-2-cl-2-l-is-at-equilibrium-in-a-closed-container-now-a-91283

question:
<div class="question">The reaction, $\mathrm{SO}_{2}+\mathrm{Cl}_{2} \longrightarrow \mathrm{SO}_{2} \mathrm{Cl}_{2}$ is
<br/>exothermic and reversible. A mixture of $\mathrm{SO}_{2}(\mathrm{~g})$, $\mathrm{Cl}_{2}(\mathrm{~g})$ and $\mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{l})$ is at equilibrium in a closed container. Now a certain quantity of extra $\mathrm{SO}_{2}$ is introduced into the container, the volume remaining the same. Which of the following is/are true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The pressure inside the container will not change.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The temperature will not change.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">The temperature will increase.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The temperature will decrease.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The temperature will increase.</span> </div>

answer:
<div class="solution">Byaddition of $\mathrm{SO}_{2}$, equilibrium will shift to RHS which is exothermic. Hence temp, will increase.</div>


source_db: MarksBatch2_P3.db
title:
the-reaction-that-does-not-take-place-in-a-blast-furnace-between-900-k-to-1500-k-temperature-range-during-extraction-of-iron-is

slug:
the-reaction-that-does-not-take-place-in-a-blast-furnace-between-900-k-to-1500-k-temperature-range-during-extraction-of-iron-is-18794

question:
<div class="question">The reaction that does NOT take place in a blast furnace between $900 \mathrm{~K}$ to $1500 \mathrm{~K}$ temperature range during extraction of iron is :</div>

tags:
['Chemistry', 'General Principles and Processes of Isolation of Metals', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{Fe}_2 \mathrm{O}_3+\mathrm{CO} \rightarrow 2 \mathrm{FeO}+\mathrm{CO}_2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{FeO}+\mathrm{CO} \rightarrow \mathrm{Fe}+\mathrm{CO}_2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{C}+\mathrm{CO}_2 \rightarrow 2 \mathrm{CO}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{CaO}+\mathrm{SiO}_2 \rightarrow \mathrm{CaSiO}_3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Fe}_2 \mathrm{O}_3+\mathrm{CO} \rightarrow 2 \mathrm{FeO}+\mathrm{CO}_2$</span> </div>

answer:
<div class="solution">At $900-1500 \mathrm{~K}$ (higher temperature range in the blast furnace)<br/>Reaction which take place are:<br/>$\mathrm{C}+\mathrm{CO}_2 \longrightarrow 2 \mathrm{CO}$<br/>$\mathrm{FeO}+\mathrm{CO} \longrightarrow \mathrm{Fe}+\mathrm{CO}_2$<br/>$\mathrm{CaO}+\mathrm{SiO}_2 \longrightarrow \mathrm{CaSiO}_3$ (Slag formation)<br/>$\mathrm{Fe}_2 \mathrm{O}_3+\mathrm{CO} \longrightarrow 2 \mathrm{FeO}+\mathrm{CO}_2$ at $500-800 \mathrm{~K}$</div>


title: the-reagent-s-which-can-be-used-to-distinguish-acetophenone-from-benzophenone-is-are
slug: the-reagent-s-which-can-be-used-to-distinguish-acetophenone-from-benzophenone-is-are-60085
tags: ['Chemistry', 'Aldehydes and Ketones', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
question:
<div class="question">The reagent (s) which can be used to distinguish acetophenone from benzophenone is (are)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2,4 - dinitrophenylhydrazine</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">aqueous solution of $\mathrm{NaHSO}_{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">benedict reagent</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{I}_{2}$ and $\mathrm{NaOH}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{I}_{2}$ and $\mathrm{NaOH}$</span> </div>

answer:
<div class="solution">$\mathrm{I}_{2}$ and $\mathrm{NaOH}$ react with acetophenone $\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COCH}_{3}\right)$ to give yellow ppt. of $\mathrm{CHI}_{3}$<br/>but benzophenone $\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COC}_{6} \mathrm{H}_{5}\right)$ does not<br/>give haloform test.</div>


tags: ['Chemistry', 'Chemical Kinetics', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
the-reduction-of-peroxydisulphate-ion-by-i-ion-is-expressed-by-s-2-o-8-2-3-i-2-so-4-2-i-3-if-rate-of-disappearance-of-i-is-92-1-0-3-mol-lit-1-s-1-what

slug:
the-reduction-of-peroxydisulphate-ion-by-i-ion-is-expressed-by-s-2-o-8-2-3-i-2-so-4-2-i-3-if-rate-of-disappearance-of-i-is-92-1-0-3-mol-lit-1-s-1-what-27404

question:
<div class="question">The reduction of peroxydisulphate ion by I ion
<br/> <br/>is expressed by $\mathrm{S}_{2} \mathrm{O}_{8}^{2-}+3 \mathrm{I}^{-} \rightarrow 2 \mathrm{SO}_{4}^{2-}+\mathrm{I}_{3}^{-}$. If
<br/> <br/>rate of disappearance of $\mathrm{I}^{-}$ is $9 / 2 \times 10^{-3} \mathrm{~mol}$ lit $^{-1}$
<br/> <br/>$\mathrm{s}^{-1},$ what is the rate of formation of $2 \mathrm{SO}_{4}^{2-}$ during same time?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3 \times 10^{-3} \mathrm{~mol} \mathrm{Lit}^{-1} \mathrm{~s}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \times 10^{-3} \mathrm{~mol} \mathrm{Lit}^{-1} \mathrm{~s}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10^{-3} \mathrm{~mol} \mathrm{Lit}^{-1} \mathrm{~s}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \times 10^{-3} \mathrm{~mol} \mathrm{Lit}^{-1} \mathrm{~s}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 \times 10^{-3} \mathrm{~mol} \mathrm{Lit}^{-1} \mathrm{~s}^{-1}$</span> </div>

answer:
<div class="solution">$\frac{1 d\left[\mathrm{SO}_{4}^{2-}\right]}{2}=\frac{1}{3}\left(-\frac{\mathrm{d}\left(\mathrm{I}^{-}\right)}{d t}\right)$
<br/> <br/>$\frac{1}{2} \times \frac{d\left[\mathrm{SO}_{4}^{2-}\right]}{d t}=\frac{1}{3} \times \frac{9}{2} \times 10^{-3}$
<br/> <br/>$\therefore \quad \frac{d\left[\mathrm{SO}_{4}^{2-}\right]}{d t}=3 \times 10^{-3} \mathrm{~mol} \mathrm{Lit}^{-1} \mathrm{~s}^{-1}$</div>


title: the-reflection-of-the-point-413-in-the-line-5-xy60-is
slug: the-reflection-of-the-point-413-in-the-line-5-xy60-is-36419
question: <div class="question">The reflection of the point \((4,-13)\) in the line \(5 x+y+6=0\), is</div>
tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\((-1,-14)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\((3,4)\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\((1,2)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\((-4,13)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((-1,-14)\)</span> </div>

answer:
<div class="solution">Let \(\mathrm{Q}(a, b)\) be the reflection of \(\mathrm{P}(4,-13)\) in the line \(5 x+y+6=0\)<br/>Then the mid-point \(R\left(\frac{a+4}{2}, \frac{b-13}{2}\right)\) lies on<br/>\(\begin{aligned}<br/>&amp; 5 x+y+6=0 \\<br/>&amp; \therefore 5\left(\frac{a+4}{2}\right)+\frac{b-13}{2}+6=0 \\<br/>&amp; \Rightarrow 5 a+b+19=0 \quad ...(i)<br/>\end{aligned}\)<br/>Also \(\mathrm{PQ}\) is perpendicular to \(5 x+y+6=0\)<br/>Therefore \(\frac{b+13}{a-4} \times\left(-\frac{5}{1}\right)=-1\)<br/>\(\Rightarrow a-5 b-69=0 \quad ...(ii)\)<br/>Solving (i) and (ii), we get \(a=-1, b=-14\)</div>


tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
title:
the-resistance-of-a-wire-at-20-c-is-20-and-500-c-is-60-at-which-temperature-its-resistance-will-be-25

slug:
the-resistance-of-a-wire-at-20-c-is-20-and-500-c-is-60-at-which-temperature-its-resistance-will-be-25-29015

question:
<div class="question">The resistance of a wire at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>500</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math>. At which temperature, its resistance will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo><mi mathvariant="normal">Ω</mi><mo> </mo><mo>?</mo></math></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>70</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math></span> </div>

answer:
<div class="solution"><p>We have, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>t</mi></msub><mo>=</mo><msub><mi>R</mi><mn>0</mn></msub><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>α</mi><mi>t</mi></mrow></mfenced></math></p><p>Here, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo>=</mo><msub><mi>R</mi><mn>0</mn></msub><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>20</mn><mi>α</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo>=</mo><msub><mi>R</mi><mn>0</mn></msub><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>500</mn><mi>α</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo> </mo><mo> </mo><mfrac><mn>60</mn><mn>20</mn></mfrac><mo>=</mo><mfrac><mrow><msub><mi>R</mi><mn>0</mn></msub><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>500</mn><mi>α</mi></mrow></mfenced></mrow><mrow><msub><mi>R</mi><mn>0</mn></msub><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>20</mn><mi>α</mi></mrow></mfenced></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mn>500</mn><mo> </mo><mi>α</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>20</mn><mo> </mo><mi>α</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>3</mn><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>20</mn><mo> </mo><mi>α</mi></mrow></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mn>500</mn><mo> </mo><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>3</mn><mo>+</mo><mn>60</mn><mi>α</mi><mo>=</mo><mn>1</mn><mo>+</mo><mn>500</mn><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>440</mn><mo> </mo><mi>α</mi><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>α</mi><mo>=</mo><mfrac><mn>2</mn><mn>440</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>220</mn></mfrac><mi>I</mi><mo>°</mo><mi mathvariant="normal">C</mi></math></p><p>Also given <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>t</mi></msub><mo>=</mo><mn>25</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi><mo>,</mo><mo> </mo><mi>t</mi><mo>=</mo><mo> </mo><mo>?</mo></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>20</mn><mn>25</mn></mfrac><mo>=</mo><mfrac><mrow><msub><mi>R</mi><mn>0</mn></msub><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>20</mn><mn>220</mn></mfrac></mrow></mfenced></mrow><mrow><msub><mi>R</mi><mn>0</mn></msub><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>t</mi><mn>220</mn></mfrac></mrow></mfenced></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>t</mi><mn>220</mn></mfrac></mrow></mfenced><mo>=</mo><mn>5</mn><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>20</mn><mn>220</mn></mfrac></mrow></mfenced></math></p><p>On solving, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>80</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math></p></div>


tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">125</span> </div>
source_db: MarksBatch2_P3.db
title:
the-resistance-of-a-wire-is-5-its-new-resistance-in-ohm-if-stretched-to-5-times-of-its-original-length-will-be-1

slug:
the-resistance-of-a-wire-is-5-its-new-resistance-in-ohm-if-stretched-to-5-times-of-its-original-length-will-be-1-53049

question:
<div class="question">The resistance of a wire is $5 \Omega$. It's new resistance in ohm if stretched to 5 times of its original length will be :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">625</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">125</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">25</span> </li> </ul>

answer:
<div class="solution"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/AHskDoEY5sab6S_juKuP1x9Cv9AcAlJ_pFWfMJGRGiM.original.fullsize.png"/><br/><br/>Let resistance of a wire $R$ and length 1 .<br/>$\mathrm{R}=\frac{\rho \ell}{\mathrm{A}}=5 \Omega$<br/>$\therefore \quad$ Volume of wire is constant in stretching<br/>$V_i=V_f \Rightarrow A_i \ell_i=A_f \ell_f$<br/>$\mathrm{A} \ell=\mathrm{A}^{\prime}(5 \ell) \Rightarrow \mathrm{A}^{\prime}=\frac{\mathrm{A}}{5}$<br/>$R_f=\frac{\rho \ell_f}{A_f}=\frac{\rho(5 \ell)}{\left(\frac{A}{5}\right)}=25\left(\frac{\rho \ell}{A}\right)=25 \times 5=125 \Omega$</div>


tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
title:
the-resistance-of-a-wire-is-r-it-is-bent-at-the-middle-by-18-0-and-both-the-ends-are-twisted-together-to-make-a-shorter-wire-the-resistance-of-the-new-1

slug:
the-resistance-of-a-wire-is-r-it-is-bent-at-the-middle-by-18-0-and-both-the-ends-are-twisted-together-to-make-a-shorter-wire-the-resistance-of-the-new-1-27553

question:
<div class="question">The resistance of a wire is $R$. It is bent at the middle by $180^{\circ}$ and both the ends are twisted together to make a shorter wire. The resistance of the new wire is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 R$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$R / 2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$R / 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$R / 8$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$R / 4$</span> </div>

answer:
<div class="solution">Resistance of wire $(R)=\rho \frac{l}{A}$ If wire is bent in the middle then
<br/> <br/>$$
<br/> <br/>l^{\prime}=\frac{l}{2}, A^{\prime}=2 A
<br/> <br/>$$
<br/> <br/>$\therefore$ New resistance, $\mathrm{R}^{\prime}=\rho \frac{l^{\prime}}{\mathrm{A}^{\prime}}=\frac{\rho \frac{l}{2}}{2 A}=$
<br/> <br/>$\frac{\rho l}{4 A}=\frac{R}{4}$</div>


title: the-rms-value-of-potential-difference-v-shown-in-the-figure-is-1
slug: the-rms-value-of-potential-difference-v-shown-in-the-figure-is-1-49911
tags: ['Physics', 'Alternating Current', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
question:
<div class="question">The r.m.s. value of potential difference $\mathrm{V}$ shown in the figure is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/5QDiyYGrdWNO8gADy-u0-3igbVsJRCuRL2hTyLf0ufk.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{V}_{0}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{V}_{0} / \sqrt{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{V}_{0} / 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{V}_{0} / \sqrt{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{V}_{0} / \sqrt{2}$</span> </div>

answer:
<div class="solution">$\mathrm{V}_{\mathrm{rms}}=\sqrt{\frac{(\mathrm{T} / 2) \mathrm{V}_{0}{ }^{2}+0}{\mathrm{~T}}}=\frac{\mathrm{V}_{0}}{\sqrt{2}}$.</div>


title: the-root-of-the-equation-2-1-i-x-2-4-2-i-x-5-3-i-0-which-has-greater-modulus-is
slug: the-root-of-the-equation-2-1-i-x-2-4-2-i-x-5-3-i-0-which-has-greater-modulus-is-83743
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
question:
<div class="question">The root of the equation
<br/> <br/>$2(1+i) x^{2}-4(2-i) x-5-3 i=0$ which has
<br/> <br/>greater modulus is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{3-5 i}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{5-3 \mathrm{i}}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3-i}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">none</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3-5 i}{2}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \text { Roots } &amp;=\frac{4(2-i) \pm \sqrt{16(2-i)^{2}+8(1+i)(5+3 i)}}{4(1+i)} \\ &amp;=\frac{4-i}{1+i} \text { or } \frac{-i}{1+i}=\frac{3-5 i}{2} \text { or } \frac{-1-i}{2} \end{aligned}$</div>


title: the-roots-of-the-equation-x22-sqrt2-x10-are
slug: the-roots-of-the-equation-x22-sqrt2-x10-are-93379
question: <div class="question">The roots of the equation \(x^2-2 \sqrt{2} x+1=0\) are</div>
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Real and different</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Imaginary and different</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Real and equal</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Rational and different</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Real and different</span> </div>

answer:
<div class="solution">The discriminant of the equation \((-2 \sqrt{2})^2-4(1)(1)=8-4=4 &gt; 0\) and \(a\) perfect square, so roots are real and different but we can't say that roots are rational because coefficients are not rational therefore.<br/>\(\frac{\sqrt{2 \sqrt{2} \pm \sqrt{(2 \sqrt{2})^2-4}}}{2}=\frac{2 \sqrt{2} \pm 2}{2}=\sqrt{2} \pm 1\)<br/>this is irrational \(\therefore\) the roots are real and different.</div>


title: the-roots-of-the-given-equation-p-q-x-2-q-r-x-r-p-0-are-1
slug: the-roots-of-the-given-equation-p-q-x-2-q-r-x-r-p-0-are-1-14680
question: <div class="question">The roots of the given equation $(p-q) x^2+(q-r) x+(r-p)=0$ are :</div>
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{p-q}{r-p}, 1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{q-r}{p-q}, 1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{r}-\mathrm{p}}{\mathrm{p}-\mathrm{q}}, 1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{r}-\mathrm{p}}{\mathrm{p}-\mathrm{q}}, 1$</span> </div>

answer:
<div class="solution">Given equation is<br/>$(p-q) x^2+(q-r) x+(r-p)=0$<br/>By using formula for finding the roots viz: $\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$, we get<br/>$\begin{aligned} &amp; x=\frac{(r-q) \pm \sqrt{(q-r)^2-4(r-p)(p-q)}}{2(p-q)} \\ &amp; \Rightarrow x=\frac{(r-q) \pm(q+r-2 p)}{2(p-q)}=\frac{r-p}{p-q}, 1\end{aligned}$</div>


title: the-roots-of-the-given-equation-p-q-x-2-q-r-x-r-p-0-are
slug: the-roots-of-the-given-equation-p-q-x-2-q-r-x-r-p-0-are-32803
question: <div class="question">The roots of the given equation $(p-q) x^2+(q-r) x+(r-p)=0$ are :</div>
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{p}-\mathrm{q}}{\mathrm{r}-\mathrm{p}}, 1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{q}-\mathrm{r}}{\mathrm{p}-\mathrm{q}}, 1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{r}-\mathrm{p}}{\mathrm{p}-\mathrm{q}}, 1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{r}-\mathrm{p}}{\mathrm{p}-\mathrm{q}}, 1$</span> </div>

answer:
<div class="solution">Given equation is<br/>$(p-q) x^2+(q-r) x+(r-p)=0$<br/>By using formula for finding the root viz: $\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$, we get<br/>$x=\frac{(r-q) \pm \sqrt{(q-r)^2-4(r-p)(p-q)}}{2(p-q)}$<br/>$\Rightarrow \quad x=\frac{(r-q) \pm(q+r-2 p)}{2(p-q)}=\frac{r-p}{p-q}, 1$</div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2009']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P3.db
title:
the-rotation-of-the-earth-having-radius-r-about-its-axis-speeds-upto-a-value-such-that-a-man-at-latitude-angle-6-0-feels-weightless-the-duration-of-th

slug:
the-rotation-of-the-earth-having-radius-r-about-its-axis-speeds-upto-a-value-such-that-a-man-at-latitude-angle-6-0-feels-weightless-the-duration-of-th-92751

question:
<div class="question">The rotation of the earth having radius $\mathrm{R}$ about its axis speeds upto a value such that a man at latitude angle $60^{\circ}$ feels weightless. The duration of the day in such case will be:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$8 \pi \sqrt{\frac{\mathrm{R}}{\mathrm{g}}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$8 \pi \sqrt{\frac{g}{R}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\pi \sqrt{\frac{\mathrm{R}}{\mathrm{g}}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \pi \sqrt{\frac{\mathrm{g}}{\mathrm{R}}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\pi \sqrt{\frac{\mathrm{R}}{\mathrm{g}}}$</span> </div>


title: the-screening-effect-of-d-electrons-is
slug: the-screening-effect-of-d-electrons-is-98071
question: <div class="question">The screening effect of $d$ -electrons is</div>
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Equal to $p$ -electrons</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Much more than $p$ -electrons</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Same as $f$ -electrons</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Less than $p$ -electrons.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Less than $p$ -electrons.</span> </div>

answer:
<div class="solution">The screening effect of inner electron of the nucleus causes the decrease in ionization potential, therefore the order of the screening effect is
<br/> <br/>$$
<br/> <br/>f<d></d> <br/>$$
<br/> <br/>Hence, the screening effect of $d$ -electron is less than $p$ -electron.</div>


title: the-secondary-structure-of-a-protein-refers-to-1
slug: the-secondary-structure-of-a-protein-refers-to-1-86098
question: <div class="question">The secondary structure of a protein refers to</div>
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">fixed configuration of the polypeptide backbone</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\alpha$ -helical backbone</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">hydrophobic interactions</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">sequence of $\alpha$ -amino acids.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\alpha$ -helical backbone</span> </div>

answer:
<div class="solution">The secondary structure of a protein refers to the shape in which a long peptide chain can exist. There are two different conformations of the peptide linkage present in protein are $\alpha$ -helix and $\beta$ -conformation. The $\alpha$ -helix always has a right handed arrangement. In $\beta$ -conformation all peptide chains are streched out to nearly maximum extension and then laid side by side held together by intermolecular hydrogen bonds. The structure resembles the pleated folds of drapery and therefore is known as $\beta$ -pleated shelt.</div>


title: the-set-a-b-b-a-is-equal-to
slug: the-set-a-b-b-a-is-equal-to-77638
question: <div class="question">The set $(A \backslash B) \cup(B \backslash A)$ is equal to]</div>
tags: ['Mathematics', 'Sets and Relations', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$[A \backslash(A \cap B)] \cap[B \backslash(A \cap B)]$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(A \cup B) \backslash(A \cap B)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$A \backslash(A \cap B)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\overrightarrow{A \cap B} \backslash A \cup B$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(A \cup B) \backslash(A \cap B)$</span> </div>

answer:
<div class="solution">Given set can be written as $(A-B) \cup(B-A)=(A \cup B)-(A \cap B)$
<br/> <br/>(By definition of symmetric difference) Hence, $(\mathrm{A} \backslash \mathrm{B}) \cup(\mathrm{B} \backslash \mathrm{A})=(\mathrm{A} \cup \mathrm{B}) \backslash(\mathrm{A} \cap \mathrm{B})$</div>


title: the-set-of-all-real-x-satisfying-the-inequality-4-x-3-x-0-is
slug: the-set-of-all-real-x-satisfying-the-inequality-4-x-3-x-0-is-86166
tags: ['Mathematics', 'Basic of Mathematics', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
question:
<div class="question">The set of all real $x$ satisfying the inequality
<br/> <br/>$\frac{3-|x|}{4-|x|} \geq 0,$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">[-3,3]$\cup(-\infty,-4) \cup(4, \infty)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(-\infty,-4) \cup(4, \infty)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-\infty,-3) \cup(4, \infty)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(-\infty,-3) \cup(3, \infty)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">[-3,3]$\cup(-\infty,-4) \cup(4, \infty)$</span> </div>

answer:
<div class="solution">Given, $\frac{3-|x|}{4-|x|} \geq 0$
<br/> <br/>$\Rightarrow 3-|x| \leq 0$ and $4-|x|&lt;0$
<br/> <br/>or $3-|x| \geq 0$ and $4-|x|&lt;0$
<br/> <br/>$\Rightarrow|x| \geq 3$ and $|x|&gt;4$
<br/> <br/>or $|x| \leq 3$ and $|x|&lt;4$
<br/> <br/>$\Rightarrow|x|&gt;4$ or $|x| \leq 3$
<br/> <br/>$\Rightarrow x \in(-\infty,-4) \cup[-3,3] \cup(4, \infty)$</div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P3.db
title:
the-set-of-all-values-of-a-for-which-the-function-f-x-a-2-3-a-2-cos-2-x-4-sin-2-x-4-a-1-x-sin-1-does-not-possess-critical-points-is

slug:
the-set-of-all-values-of-a-for-which-the-function-f-x-a-2-3-a-2-cos-2-x-4-sin-2-x-4-a-1-x-sin-1-does-not-possess-critical-points-is-77177

question:
<div class="question">The set of all values of a for which the function $f(x)=\left(a^{2}-3 a+2\right)\left(\cos ^{2} x / 4-\sin ^{2} x / 4\right)+(a-1) x+$
<br/> <br/>$\sin 1$ does not possess critical points is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$[1, \infty)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">(0,1)$\cup(1,4)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(-2,4)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">(1,3)$\cup(3,5)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">(0,1)$\cup(1,4)$</span> </div>

answer:
<div class="solution">\(f(x)=\left(a^{2}=-3 a+2\right)\left(\cos ^{2} x / 4-\sin ^{2} x / 4\right)+(a-1) x+\sin 1\)
<br/> <br/>\(\Rightarrow \mathrm{f}(\mathrm{x})=(\mathrm{a}-1)(\mathrm{a}-2) \cos \mathrm{x} / 2+(\mathrm{a}-1) \mathrm{x}+\sin 1\)
<br/> <br/>\(\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=-\frac{1}{2}(\mathrm{a}-1)(\mathrm{a}-2) \sin \frac{\mathrm{x}}{2}+(\mathrm{a}-1)\)
<br/> <br/>\(\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=(\mathrm{a}-1)\left[1-\frac{(\mathrm{a}-2)}{2} \sin \frac{\mathrm{x}}{2}\right]\)
<br/> <br/>If \(f(x)\) does not possess critical points, then \(f^{\prime}(x)\) = 0 for any \(x \in R\)
<br/> <br/>\(\Rightarrow(a-1)\left[1-\frac{(a-2)}{2} \sin \frac{x}{2}\right]\) = 0 for any \(x \in R\)
<br/> <br/>\(\Rightarrow \mathrm{a}=1\) and \(1-\left(\frac{\mathrm{a}-2}{2}\right) \sin \frac{\mathrm{x}}{2}=0\)
<br/> <br/>must not have any solution in \(\mathrm{R}\).
<br/> <br/>\(\Rightarrow \mathrm{a}=1\) and \(\sin \frac{\mathrm{x}}{2}=\frac{2}{\mathrm{a}-2}\) is not solvable in \(\mathrm{R}\).
<br/> <br/>\(\Rightarrow \mathrm{a} =  1\) and \(\left|\frac{2}{\mathrm{a}-2}\right|&gt;1\left[\right.\) For \(\mathrm{a}=2, \mathrm{f}(\mathrm{x})=\mathrm{x}+\sin 1 \therefore \mathrm{f}^{\prime}(\mathrm{x})=1\)
<br/> <br/>\(\Rightarrow \mathrm{a}=1\) and \(0&lt;\mathrm{a}&lt;4 \Rightarrow \mathrm{a} \in(0,1) \cup(1,4)\).</div>


title: the-set-of-points-of-discontinuity-of-the-function-1-log-mathrmx-is
slug: the-set-of-points-of-discontinuity-of-the-function-1-log-mathrmx-is-37812
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P3.db
question:
<div class="question">The set of points of discontinuity of the function \(1 / \log |\mathrm{x}|\) is -</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\{-1,0,1\}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\{0\}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\{0,1\}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\{-1,0,1\}\)</span> </div>

answer:
<div class="solution">Let \(f(x)=\frac{1}{\log |x|}\)<br/>The points of disconinuity of \(f(x)\) are those points where \(f(x)\) is undefined or infinite. It is undefined when \(x=0\) and is infinite when \(\log |x|=0,|x|=1\), i.e. \(x= \pm 1\).<br/>\(\therefore \quad\) Set of points of discontinuity \(=\{-1,0,1\}\).</div>


title: the-shortest-distance-between-the-lines-2-x-3-3-y-2-1-z-1-and-2-x-3-1-y-6-3-z-5-is
slug: the-shortest-distance-between-the-lines-2-x-3-3-y-2-1-z-1-and-2-x-3-1-y-6-3-z-5-is-12085
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The shortest distance between the lines $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}$ and $\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}$ is :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{18}{\sqrt{5}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{22}{3 \sqrt{5}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> $\frac{46}{3 \sqrt{5}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$6 \sqrt{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{18}{\sqrt{5}}$</span> </div>

answer:
<div class="solution">Give lines are $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}$ and $\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}$<br/>Lines passes through the points<br/>$\vec{a}_1=(3,2,1) \text { and } \vec{a}_2=(-3,6,5) \text {, }$<br/>$\vec{b}_1=2 \hat{i}+3 \hat{j}-\hat{k}$<br/>$\vec{b}_1=2 \hat{i}+\hat{j}-3 \hat{k}, \vec{a}_2-\vec{a}_1=6 \hat{i}-4 j-4 \hat{k}$<br/>Shortest distance $=\frac{\left|\left(\vec{a}_2-\vec{a}_1\right)\left(\vec{b}_1 \times \vec{b}_2\right)\right|}{\left|\left(\vec{b}_1 \times \vec{b}_2\right)\right|}$<br/>$\vec{b}_1 \times \vec{b}_2=\left|\begin{array}{ccc}\hat{i} &amp; \hat{j} &amp; \hat{k} \\ 2 &amp; 3 &amp; -1 \\ 2 &amp; 1 &amp; 3\end{array}\right|=10 \hat{i}-8 \hat{j}-4 \hat{k}$<br/>$\left(\vec{a}_2-\vec{a}_1\right) \cdot\left(\vec{b}_1 \times \vec{b}_2\right)=60+32+16=108$<br/>$\left|\vec{b}_1 \times \vec{b}_2\right|=\sqrt{100+64+16}=\sqrt{180}$<br/>$S . D=\frac{108}{\sqrt{180}}=\frac{108}{6 \sqrt{5}}=\frac{18}{\sqrt{5}}$</div>


title: the-shortest-distance-between-the-lines-x-y-2-6-z-6-and-x-1-2-y-12-z-is-2
slug: the-shortest-distance-between-the-lines-x-y-2-6-z-6-and-x-1-2-y-12-z-is-2-22702
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The shortest distance between the lines $\mathrm{x}=\mathrm{y}+$ $2=6 z-6$ and $x+1=2 y=-12 z$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3}{2}$</span> </li> </ul>

answer:
<div class="solution">The lines are $\frac{x}{6}=\frac{y+2}{6}=\frac{z-1}{1}$<br/>and $\frac{x+1}{12}=\frac{y}{6}=\frac{z}{-1}$<br/>Here,<br/>$\begin{aligned}<br/>&amp; \vec{a}_1=-2 \hat{j}+\hat{k}, b_1+6 \hat{i}+6 \hat{j}+\hat{k}, \vec{a}_2=-\hat{i}, \\<br/>&amp; \vec{b}_2=12 \hat{i}+6 \hat{j}-\hat{k} \\<br/>&amp; \vec{b}_1 \times \vec{b}_2=\left|\begin{array}{ccc}<br/>\hat{i} &amp; \hat{j} &amp; k \\<br/>6 &amp; 6 &amp; 1 \\<br/>12 &amp; 6 &amp; -1<br/>\end{array}\right|=-12 \hat{i}+18 \hat{j}-36 \hat{k} \\<br/>&amp; \text { Shortest distance }=\frac{\left|\left(\vec{a}_2-\vec{a}_1\right) \cdot\left(\vec{b}_1-\vec{b}_2\right)\right|}{\left|\vec{b}_1 \times \vec{b}_2\right|}<br/>\end{aligned}$<br/>$\begin{aligned} &amp; =\frac{|(-\hat{i}+2 \hat{j}-\hat{k}) \cdot(-12 i+18 \hat{j}-36 \hat{k})|}{\sqrt{(-12)^2+(18)^2+(-36)^2}} \\ &amp; =\frac{|+12+36+36|}{\sqrt{1764}}=\frac{84}{42}=2\end{aligned}$</div>


title: the-shortest-distance-between-the-parabolas-y-2-4-x-and-y-2-2-x-6-is
slug: the-shortest-distance-between-the-parabolas-y-2-4-x-and-y-2-2-x-6-is-26382
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
question:
<div class="question">The shortest distance between the parabolas <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>6</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>5</mn></msqrt></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>5</mn></msqrt></math></span> </div>

answer:
<div class="solution"><p>Shortest distance between two curves occured along the common normal.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mi>x</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>,</mo><mn>2</mn><mi>m</mi></mrow></mfenced></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mi>m</mi><mi>x</mi><mo>-</mo><mn>2</mn><mi>m</mi><mo>-</mo><msup><mi>m</mi><mn>3</mn></msup><mo>=</mo><mn>0</mn></math></p><p>Normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mn>3</mn><mo>,</mo><mi>m</mi></mrow></mfenced></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mi>m</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mi>m</mi><mo>-</mo><mfrac><msup><mi>m</mi><mn>3</mn></msup><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math></p><p>Both normals are same, if</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mi>m</mi><mo>-</mo><msup><mi>m</mi><mn>3</mn></msup><mo>=</mo><mo>-</mo><mn>4</mn><mi>m</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>m</mi><mn>3</mn></msup><mo>⇒</mo><mi>m</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>2</mn></math></p><p>So, points will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mn>4</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>5</mn><mo>,</mo><mn>2</mn><mo>)</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>4</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>5</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>)</mo></math>.</p><p>Hence, shortest distance will be </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mo>(</mo><mn>1</mn><mo>+</mo><mn>4</mn><mo>)</mo></msqrt><mo>=</mo><msqrt><mn>5</mn></msqrt></math> unit.</p></div>


title: the-significant-result-deduced-from-the-rutherfords-scattering-experiment-is-that
slug: the-significant-result-deduced-from-the-rutherfords-scattering-experiment-is-that-45307
tags: ['Physics', 'Atomic Physics', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
question:
<div class="question">The significant result deduced from the Rutherford's scattering experiment is that</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">whole of the positive charge is concentrated at the centre of atom</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">there are neutrons inside the nucleus</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\alpha$ - particles are helium nuclei</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">electrons are embedded in the atom</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">whole of the positive charge is concentrated at the centre of atom</span> </div>

answer:
<div class="solution">The significant result deduced from the Rutherford's scatter ing is that whole of the positive charge is concentrated at the centre of atom i.e. nucleus.</div>


title: the-slope-of-the-tangent-to-the-curve-x-3-t-2-1-y-t-3-1-at-x-1-is-2
slug: the-slope-of-the-tangent-to-the-curve-x-3-t-2-1-y-t-3-1-at-x-1-is-2-46445
tags: ['Mathematics', 'Differentiation', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The slope of the tangent to the curve $x=3 t^2+1$, $y=t^3-1$ at $x=1$ is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>-2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\infty$</span> </li> </ul>

answer:
<div class="solution">Given curve is $x=3 t^2+1$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/i96DNHHTo_FW52n7W2lHdlYYGcu7QfMaYWl6ffqxwB0.original.fullsize.png"/><br/><br/>Second curve is $y=t^3-1$<img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Amt76KEG_dlbSyhjkiFOkKWDMGfdWOuylzm_d0Px6vA.original.fullsize.png"/><br/><br/>$\therefore \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=3 t^2 \times \frac{1}{6 t}=\frac{t}{2}$<br/>But from (i) when $x=1$<br/>we have $1=3 t^2+1 \Rightarrow 3 t^2=0 \Rightarrow t=0$<br/>$\therefore$ When $\mathrm{x}=1$ then $\mathrm{t}=0 \quad \therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\mathbf{0}$<br/>Hence, slope of the tangent to the curve $=0$</div>


title: the-slope-of-the-tangent-to-the-curve-y-e-x-cos-x-is-minimum-at-x-0-a-2-then-the-value-of-is
slug: the-slope-of-the-tangent-to-the-curve-y-e-x-cos-x-is-minimum-at-x-0-a-2-then-the-value-of-is-50893
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$\pi$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The slope of the tangent to the curve $y=e^{x}$ cos $x$ is minimum at $x=\alpha, 0 \leq a \leq 2 \pi$, then the value of $\alpha$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\underline{0}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\pi$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \pi$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 \pi 2$</span> </li> </ul>

answer:
<div class="solution">Let $m$ be the slope of the tangent to the curve
<br/> <br/>$y=e^{x} \cos x$
<br/> <br/>Then, $m=\frac{d y}{d x}=e^{x}(\cos x-\sin x)$
<br/> <br/>Diff. w.r.t 'x'
<br/> <br/>$$
<br/> <br/>\begin{array}{r}
<br/> <br/>\Rightarrow \frac{\mathrm{dm}}{\mathrm{dx}}=\mathrm{e}^{\mathrm{x}}(\cos \mathrm{x}-\sin \mathrm{x})+\mathrm{e}^{\mathrm{x}}(-\sin \mathrm{x}-\cos \mathrm{x}) \\
<br/> <br/>=-2 \mathrm{e}^{\mathrm{x}} \sin \mathrm{x}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>and $\frac{\mathrm{d}^{2} \mathrm{~m}}{\mathrm{dx}^{2}}=-2 \mathrm{e}^{\mathrm{x}}(\sin \mathrm{x}+\cos \mathrm{x})$
<br/> <br/>Put $\frac{\mathrm{dm}}{\mathrm{dx}}=0 \Rightarrow \sin \mathrm{x}=0 \Rightarrow \mathrm{x}=0, \pi, 2 \pi$
<br/> <br/>Clearly, $\frac{\mathrm{d}^{2} \mathrm{~m}}{\mathrm{dx}^{2}}&gt;0$ for $\mathrm{x}=\pi$
<br/> <br/>Thus, $y$ is minimum at $x=\pi$.
<br/> <br/>Hence the value of $\alpha=\pi$</div>


title: the-slope-of-the-tangent-to-the-hyperbola-2-x23-y26-at-32-is
slug: the-slope-of-the-tangent-to-the-hyperbola-2-x23-y26-at-32-is-58371
tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The slope of the tangent to the hyperbola \(2 x^2-3 y^2=6\) at \((3,2)\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Differentiating the given equation of the curve<br/>\(\begin{aligned}<br/>&amp; 4 x-6 y \cdot(\mathrm{dy} / \mathrm{dx})=0 \quad \therefore \mathrm{dy} / \mathrm{dx}=2 \mathrm{x} / 3 \mathrm{y} \\<br/>&amp; \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{(3,2)}=\frac{2}{3} \cdot \frac{3}{2}=1<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
title:
the-smallest-positive-integral-value-of-n-such-that-1-s-i-n-8-i-c-o-s-8-1-s-i-n-8-i-c-o-s-8-n-is-purely-imaginary-is-equal-to

slug:
the-smallest-positive-integral-value-of-n-such-that-1-s-i-n-8-i-c-o-s-8-1-s-i-n-8-i-c-o-s-8-n-is-purely-imaginary-is-equal-to-69266

question:
<div class="question">The smallest positive integral value of $n$ such that 
<br/>$\left[\frac{1+\sin \frac{\pi}{8}+i \cos \frac{\pi}{8}}{1+\sin \frac{\pi}{8}-i \cos \frac{\pi}{8}}\right]^n$ is purely imaginary, is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>$4$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$3$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$2$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$8$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$4$
<br/></span> </div>

answer:
<div class="solution">$\left[\frac{1+\sin \frac{\pi}{8}+i \cos \frac{\pi}{8}}{1+\sin \frac{\pi}{8}-i \cos \frac{\pi}{8}}\right]^n=\left[\frac{1+\cos \alpha+i \sin \alpha}{1+\cos \alpha-i \sin \alpha}\right]^n$
<br/>$\left[\because \alpha=\frac{\pi}{2}-\frac{\pi}{8}\right]$
<br/>$=\left[\frac{2 \cos ^2 \frac{\alpha}{2}+2 i \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}}{2 \cos ^2 \frac{\alpha}{2}-2 i \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}}\right]^n$
<br/>$=\left[\frac{\cos \frac{\alpha}{2}+i \sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2}-i \sin \frac{\alpha}{2}}\right]_{-}^n=\left(e^{2 i \frac{\alpha}{2}}\right]^n=e^{i n \alpha}$
<br/>$=e^{i n\left(\frac{3 \pi}{8}\right)}=\cos \frac{3 n \pi}{8}+i \sin \frac{3 n \pi}{8}$
<br/>For $n=4$ we get imaginary part.</div>


title: the-solubility-of-silver-bromide-in-hypo-solution-involves-the-formation-of
slug: the-solubility-of-silver-bromide-in-hypo-solution-involves-the-formation-of-94348
tags: ['Chemistry', 'Ionic Equilibrium', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
question:
<div class="question">The solubility of silver bromide in hypo solution involves the formation of:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Ag}_{2} \mathrm{SO}_{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{Ag}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Ag}^{+}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{NH}_{4}{ }^{+}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Ag}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}$</span> </div>

answer:
<div class="solution">Hypo solution is $\mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}$ solution which is used in photography for fixing films \&amp; prints. Photographic emulsions are made of AgBr. After developing, the film is put into hypo solution. This forms soluble complex with $\mathrm{Ag}$.
<br/> <br/>$\mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}+\mathrm{Ag} \mathrm{Br} \rightarrow \mathrm{Ag}_{2} \mathrm{~S}_{2} \mathrm{O}_{3} \longrightarrow \mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3} }{\mathrm{Na}_{5}\left[\mathrm{Ag}\left(\mathrm{S}_{2} \mathrm{O}_{3}\right)_{3}\right]$</div>


title: the-solution-of-2-cos-x-1-3-2-cos-x-0-in-the-interval-0-x-2-is
slug: the-solution-of-2-cos-x-1-3-2-cos-x-0-in-the-interval-0-x-2-is-57298
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
question:
<div class="question">The solution of $(2 \cos x-1)(3+2 \cos x)=0$ in the interval $0 \leq x \leq 2 \pi$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{3}, \frac{5 \pi}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{3}, \frac{5 \pi}{3}, \cos ^{-1}\left(-\frac{3}{2}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi}{3}, \frac{5 \pi}{3}$</span> </div>

answer:
<div class="solution">We have $(2 \cos x-1)(3+2 \cos x)=0$
<br/> <br/>If $2 \cos x-1=0,$ then $\cos x=\frac{1}{2}$
<br/> <br/>$\therefore x=\pi / 3,5 \pi / 3$
<br/> <br/>If $3+2 \cos x=0,$ the $\cos x=-3 / 2$
<br/> <br/>which is not possible.</div>


title: the-solution-of-d-y-d-x-cos-x-y-sin-x-y-is
slug: the-solution-of-d-y-d-x-cos-x-y-sin-x-y-is-78283
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
question:
<div class="question">The solution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced></math>, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced close="]" open="["><mrow><mn>1</mn><mo>+</mo><mi>tan</mi><mfenced><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac></mfenced></mrow></mfenced><mo>+</mo><mi>C</mi><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced close="]" open="["><mrow><mn>1</mn><mo>+</mo><mi>tan</mi><mfenced><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac></mfenced></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced close="]" open="["><mrow><mn>1</mn><mo>-</mo><mi>tan</mi><mfenced><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac></mfenced></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced close="]" open="["><mrow><mn>1</mn><mo>+</mo><mi>tan</mi><mfenced><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac></mfenced></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></span> </div>

answer:
<div class="solution"><p>Put <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi>v</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></math></p><p>Therefore, the differential equation reduces to</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>cos</mi><mi>v</mi></mrow></mfenced><mo>+</mo><mi>sin</mi><mi>v</mi><mo>=</mo><mn>2</mn><msup><mi>cos</mi><mn>2</mn></msup><mfrac><mi>v</mi><mn>2</mn></mfrac><mo>+</mo><mn>2</mn><mi>sin</mi><mfrac><mi>v</mi><mn>2</mn></mfrac><mi>cos</mi><mfrac><mi>v</mi><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><msup><mi>cos</mi><mn>2</mn></msup><mfrac><mi>v</mi><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mi>tan</mi><mi>v</mi></mrow><mn>2</mn></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo>∫</mo><mfrac><mrow><msup><mi>sec</mi><mn>2</mn></msup><mfrac><mi>V</mi><mn>2</mn></mfrac></mrow><mrow><mn>2</mn><mfenced close="]" open="["><mrow><mn>1</mn><mo>+</mo><mi>tan</mi><mfenced><mfrac><mi>v</mi><mn>2</mn></mfrac></mfenced></mrow></mfenced></mrow></mfrac><mi>d</mi><mi>v</mi><mo>=</mo><mo>∫</mo><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>log</mi><mfenced close="]" open="["><mrow><mn>1</mn><mo>+</mo><mi>tan</mi><mfenced><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac></mfenced></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></p></div>


title: the-solution-of-differential-equation-2-x-fracd-yd-xy3-represents-a-family-of
slug: the-solution-of-differential-equation-2-x-fracd-yd-xy3-represents-a-family-of-27800
tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
question:
<div class="question">The solution of differential equation \(2 x \frac{d y}{d x}-y=3\) represents a family of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">circles</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">straight lines</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">ellipses</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">parabola</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">parabola</span> </div>

answer:
<div class="solution">We have, \(2 x \frac{d y}{d x}=y+3 \Rightarrow \frac{2}{y+3} d y=\frac{d x}{x}\) integrating, \(2 \ln (y+3)=\ln x+\ln c=\ln c x\)<br/>\(\Rightarrow \ln (y+3)^2=\ln \mathrm{cx} \Rightarrow(y+3)^2=\mathrm{cx}\)<br/>which is a family of parabolas.</div>


title: the-solution-of-the-differential-equation-d-2-y-d-x-2-sin-3-x-e-x-x-2-when-y-1-0-1-and-y-0-0-is
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
slug:
the-solution-of-the-differential-equation-d-2-y-d-x-2-sin-3-x-e-x-x-2-when-y-1-0-1-and-y-0-0-is-74906

question:
<div class="question">The solution of the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>d</mi><mn>2</mn></msup><mi>y</mi></mrow><mrow><mi>d</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>sin</mi><mn>3</mn><mi>x</mi><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>1</mn></msub><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math>, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mi>sin</mi><mn>3</mn><mi>x</mi></mrow><mn>9</mn></mfrac><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>12</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>-</mo><mn>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mi>sin</mi><mn>3</mn><mi>x</mi></mrow><mn>9</mn></mfrac><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>12</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mi>cos</mi><mn>3</mn><mi>x</mi></mrow><mn>9</mn></mfrac><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>12</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mi>sin</mi><mn>3</mn><mi>x</mi></mrow><mn>9</mn></mfrac><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>12</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>-</mo><mn>1</mn></math></span> </div>

answer:
<div class="solution"><p>Integrating the given differential equation, we have</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mi>cos</mi><mn>3</mn><mi>x</mi></mrow><mn>3</mn></mfrac><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mo>+</mo><msub><mi>C</mi><mn>1</mn></msub></math></p><p>But <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mn>1</mn></msub><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn></math></p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn><mo>+</mo><msub><mi>C</mi><mn>1</mn></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>C</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mi>cos</mi><mn>3</mn><mi>x</mi></mrow><mn>3</mn></mfrac><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p>Again integrating, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>sin</mi><mn>3</mn><mi>x</mi></mrow><mn>9</mn></mfrac><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>12</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub></math></p><p>But <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn><mo>,</mo></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>0</mn><mo>+</mo><mn>1</mn><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>C</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mn>1</mn></math></p><p>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>sin</mi><mn>3</mn><mi>x</mi></mrow><mn>9</mn></mfrac><mo>+</mo><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>12</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>x</mi><mo>-</mo><mn>1</mn></math></p><p>Hence, option (a) correct.</p></div>


title: the-solution-of-the-differential-equation-x-1-d-x-d-y-y-e-3-x-x-1-2-is
slug: the-solution-of-the-differential-equation-x-1-d-x-d-y-y-e-3-x-x-1-2-is-52941
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
question:
<div class="question">The solution of the differential equation
<br/> <br/>$(x+1) \frac{d y}{d x}-y=e^{3 x}(x+1)^{2}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y=(x+1) e^{3 x}+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 y=(x+1)+e^{3 x}+c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{3 y}{x+1}=e^{3 x}+c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y e^{-3 x}=3(x+1)+c$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3 y}{x+1}=e^{3 x}+c$</span> </div>

answer:
<div class="solution">The given equation is $\frac{d y}{d x}-\frac{y}{x+1}=e^{3 x}(x+1)$
<br/> <br/>I.F. $=e^{\int-\frac{1}{x+1} d x}=e^{-\log (x+1)}=\frac{1}{x+1}$
<br/> <br/>The solution is
<br/> <br/>$y\left(\frac{1}{x+1}\right)=\int e^{3 x}(x+1) \cdot \frac{1}{x+1} d x+a$
<br/> <br/>$\Rightarrow \frac{y}{x+1}=\int e^{3 x} d x+a=\frac{e^{3 x}}{3}+a$
<br/> <br/>$\Rightarrow \frac{3 y}{x+1}=e^{3 x}+c, c=3 a$</div>


title: the-solution-set-of-x-2-x-3-6-x-8-1-is
slug: the-solution-set-of-x-2-x-3-6-x-8-1-is-55617
question: <div class="question">The solution set of $(x-2)^{x^3-6 x+8}&gt;1$ is</div>
tags: ['Mathematics', 'Basic of Mathematics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(2, \infty)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(2,3) \cup(4, \infty)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(4,5) \cup(5, \infty)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(2,3) \cup(4,5)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(2,3) \cup(4, \infty)$</span> </div>

answer:
<div class="solution">Clearly $x&gt;2$. Write the given inequality<br/>as<br/>$\begin{aligned}<br/>&amp; \left(x^2-6 x+8\right) \log (x-2)&gt;0 \Leftrightarrow(x-2)(x-4) \log (x-2)&gt; \\<br/>&amp; 0 \\<br/>&amp; \Leftrightarrow(x-4) \log (x-2)&gt;0 \quad[\therefore x&gt;2]<br/>\end{aligned}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/3oxXvNf76QSQceF-Ow1dFlaYH4BOy9y9a8VtiBEtZ-E.original.fullsize.png"/><br/></div>


title: the-solution-to-the-differential-equation-d-x-d-y-f-x-y-f-x-y-2-where-f-x-is-a-given-function-is
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
slug:
the-solution-to-the-differential-equation-d-x-d-y-f-x-y-f-x-y-2-where-f-x-is-a-given-function-is-19163

question:
<div class="question">The solution to the differential equation $\frac{d y}{d x}=\frac{y f^{\prime}(x)-y^{2}}{f(x)}$ where $f(x)$ is a given function is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$f(x)=y(x+c)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f(x)=c x y$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f(x)=c(x+y)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{yf}(\mathrm{x})=\mathrm{cx}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(x)=y(x+c)$</span> </div>

answer:
<div class="solution">We have $\frac{d y}{d x}=\frac{f^{\prime}(x)}{f(x)} y-\frac{y^{2}}{f(x)}$
<br/> <br/>$\Rightarrow \frac{d y}{d x}-\frac{f^{\prime}(x)}{f(x)} y=-\frac{y^{2}}{f(x)}$
<br/> <br/>Divide by $y^{2}: y^{-2} \frac{d y}{d x}-y^{-1} \frac{f^{\prime}(x)}{f(x)}=-\frac{1}{f(x)}$
<br/> <br/>Put $y^{-1}=z \Rightarrow-y^{-2} \frac{d y}{d x}=\frac{d z}{d x}$
<br/> <br/>$-\frac{d z}{d x}-\frac{f^{\prime}(x)}{f(x)}(z)=-\frac{1}{f(x)}$
<br/> <br/>$\Rightarrow \frac{d z}{d x}+\frac{f^{\prime}(x)}{f(x)} z=\frac{1}{f(x)}$
<br/> <br/>$I F=e^{\int \frac{f^{\prime}(x)}{f(x)} d x}=e^{\log f(x)}=f(x)$
<br/> <br/>$\therefore$ The solution is $z(f(x))=\int \frac{1}{f(x)}(f(x)) d x+c$
<br/> <br/>$\Rightarrow y^{-1}(f(x))=x+c \Rightarrow f(x)=y(x+c)$</div>


title: the-solutions-of-x-y-1-d-y-d-x-are
slug: the-solutions-of-x-y-1-d-y-d-x-are-41485
question: <div class="question">The solutions of $(x+y+1) d y=d x$ are</div>
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x+y+2=C e^{y}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x+y+4=C \log y$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \log (x+y+2)=C y$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\log (x+y+2)=C-y$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x+y+2=C e^{y}$</span> </div>

answer:
<div class="solution">Putting $x+y+1=u,$ we have $d u=d x+d y$ and the given equation reduces to $u(d u-d x)=$ $d x$
<br/> <br/>$\Rightarrow \quad \frac{u d u}{u+1}=d x \quad \Rightarrow \quad u-\log (u+1)=x$
<br/> <br/>$\Rightarrow \log (x+y+2)=y+$ constant
<br/> <br/>$\Rightarrow x+y+2=C e^{y}$</div>


title: the-standard-deviation-of-5-scores-1-2-3-4-5-is-a-the-value-of-a-is
slug: the-standard-deviation-of-5-scores-1-2-3-4-5-is-a-the-value-of-a-is-39247
tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The standard deviation of 5 scores $1,2,3,4,5$ is $\sqrt{a}$. The value of ' $a$ ' is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>5</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>1</span> </li> </ul>

answer:
<div class="solution"><br/>$\begin{aligned} &amp; \text { Mean }(\bar{x})=\frac{1+2+3+4+5}{5}=3 \\ &amp; \text { S.D }=\sigma=\sqrt{\frac{1}{5}(1+4+9+16+25)-9}=\sqrt{11-9} \\ &amp; =\sqrt{2}\end{aligned}$</div>


title: the-starting-material-for-the-manufacture-of-kmno-4-is
slug: the-starting-material-for-the-manufacture-of-kmno-4-is-17847
question: <div class="question">The starting material for the manufacture of $\mathrm{KMnO}_4$ is</div>
answer: <div class="solution">Pyrolusite (It is $\mathrm{MnO}_2$ )</div>
source_db: MarksBatch2_P3.db
tags:
['Chemistry', 'General Principles and Processes of Isolation of Metals', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>pyrolusite</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>manganite</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>magnatite</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>haematite</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>pyrolusite</span> </div>


title: the-statement-p-for-any-real-numbers-x-y-if-x-y-then-2-x-a-2-y-a-when-a-z
slug: the-statement-p-for-any-real-numbers-x-y-if-x-y-then-2-x-a-2-y-a-when-a-z-62814
tags: ['Mathematics', 'Mathematical Reasoning', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
question:
<div class="question">The statement $p:$ For any real numbers $x, y$ if $x=y$, then $2 x+a=2 y+a$ when $a \in \mathrm{Z}$.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">is true</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">is false</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">its contrapositive is not true</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">is true</span> </div>

answer:
<div class="solution">We prove the statement $p$ is true by contrapositive method and by direct method.
<br/> <br/>Direct method: For any real number $x$ and $y$,
<br/> <br/>$\begin{array}{l}
<br/> <br/>x=y \\
<br/> <br/>\Rightarrow 2 x=2 y \\
<br/> <br/>\Rightarrow 2 x+a=2 y+a \text { a for some } a \in Z
<br/> <br/>\end{array}$
<br/> <br/>Contrapositive method: The contrapositive statement of $p$ is "For any real numbers $x, y$ if $2 \mathrm{x}+\mathrm{a} \neq 2 \mathrm{y}+\mathrm{a}$, where $\mathrm{a} \in \mathrm{Z}$, then $\mathrm{x} \neq \mathrm{y}$."
<br/> <br/>Given, $2 x+a \neq 2 y+a$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\Rightarrow 2 x \neq 2 y \\
<br/> <br/>\Rightarrow x \neq y
<br/> <br/>\end{array}$
<br/> <br/>Hence, the given statement is true.</div>


title: the-statement-which-is-not-correct-about-control-of-particulate-pollution-is
slug: the-statement-which-is-not-correct-about-control-of-particulate-pollution-is-17986
tags: ['Chemistry', 'Environmental Chemistry', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P3.db
question:
<div class="question">The statement which is not correct about control of particulate pollution is:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">In electrostatic precipitator, the particulates are made to acquire positive charge which are then attracted by the negative electrode and removed.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Gravity settling chamber removes larger particles from the air.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Cyclone collector removes fine particls in the diameter range $5-20$ microns.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Wet scrubbers are used to wash away all types of particulates.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">In electrostatic precipitator, the particulates are made to acquire positive charge which are then attracted by the negative electrode and removed.</span> </div>

answer:
<div class="solution">Particulates acquire negative charge and are attracted by the positive electrode.</div>


tags: ['Physics', 'Atomic Physics', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
title:
the-stopping-potential-v-0-versus-frequency-v-of-a-graph-for-photoelectric-effect-in-a-metal-from-the-graph-the-plancks-constant-h-is

slug:
the-stopping-potential-v-0-versus-frequency-v-of-a-graph-for-photoelectric-effect-in-a-metal-from-the-graph-the-plancks-constant-h-is-30604

question:
<div class="question">The stopping potential $\left(V_0\right)$ versus frequency $(v)$ of a graph for photoelectric effect in a metal. From the graph, the planck's constant $(h)$ is.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/UFPKmTwLy7iBIZh1vRP5f2ZEs3UvEx26A1Sm-ij5hc0.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>$6.60 \times 10^{-34} \mathrm{~J}-\mathrm{s}$ 
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$6.69 \times 10^{-34} \mathrm{~J}-\mathrm{s}$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$6.62 \times 10^{-34} \mathrm{~J}-\mathrm{s}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$6.63 \times 10^{-34} \mathrm{~J}-\mathrm{s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$6.60 \times 10^{-34} \mathrm{~J}-\mathrm{s}$ 
<br/></span> </div>

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>e V_0=h\left(v-v_0\right) &amp; =h v-h v_0 \\
<br/>V_0 &amp; =\left(\frac{h}{e}\right) v-\left(\frac{h}{e}\right) v_0       \quad \quad \ldots (i)
<br/>\end{aligned}
<br/>$$
<br/>From Eq. (i), it follows that the slope of the graph is $\frac{h}{e}$. Therefore,
<br/>$$
<br/>\begin{aligned}
<br/>h=e \times \text { slope } &amp; =1.6 \times 10^{-19} \times \frac{165-0}{(8-4) \times 10^{15}} \\
<br/>&amp; =\frac{1.6 \times 10^{-19} \times 16.5}{4 \times 10^{15}}=6.6 \times 10^{-34} \mathrm{~J}-\mathrm{s} .
<br/>\end{aligned}
<br/>$$</div>


title: the-structural-feature-which-distinguishes-proline-from-natural-amino-acids-1
slug: the-structural-feature-which-distinguishes-proline-from-natural-amino-acids-1-15739
tags: ['Chemistry', 'Biomolecules', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question">The structural feature which distinguishes proline from natural $\alpha$ -amino acids?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Proline is optically inactive.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Proline contains aromatic group.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Proline is a dicarboxylic acid.</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Proline is a secondary amine.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Proline is a secondary amine.</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/EVjiw_wywOmLBIoma9hdXxh2tjN1bH_um9XjbEc5cT4.original.fullsize.png"/></div>


title: the-structure-of-h-2-o-2-is-1
slug: the-structure-of-h-2-o-2-is-1-91299
question: <div class="question">The structure of $\mathrm{H}_2 \mathrm{O}_2$ is</div>
tags: ['Chemistry', 'Hydrogen', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>planar, linear
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>non-planar, linear
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>planar, non-linear
<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"> <br/>non-planar, non-linear</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>non-planar, non-linear</span> </div>

answer:
<div class="solution">Hydrogen peroxide $\left(\mathrm{H}_2 \mathrm{O}_2\right)$ is a non-planar molecule with twisted symmetrical structure. It has non-linear structure with an open book structure.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/UQ9SMbB9p4ao6qVk3JEXZAwWkaLw36PYNwbrNoZBwp8.original.fullsize.png"/><br/></div>


title: the-sum-of-all-odd-numbers-between-1-and-1000-which-are-divisible-by-3-is
slug: the-sum-of-all-odd-numbers-between-1-and-1000-which-are-divisible-by-3-is-91056
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">83667</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The sum of all odd numbers between 1 and 1000 which are divisible by 3 is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">83667</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">90000</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">83660</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">Sum of odd numbers between 1 and 1000 , which is divisible by $3=3+9+15+21+27+\ldots+999=\mathrm{S}$ (let)
<br/> <br/>$\therefore$ Let $\mathrm{n}$ be the number of terms in series and $\mathrm{a}$ is first term.
<br/> <br/>$\therefore l=a+(n-1) d$,
<br/> <br/>where $l$ is last term and $d$ is is common difference.
<br/> <br/>$\begin{array}{l}
<br/> <br/>999=3+(n-1) \times 6 \\
<br/> <br/>n-1=\frac{999-3}{6}=\frac{996}{6} \\
<br/> <br/>\Rightarrow n-1=166 \\
<br/> <br/>\Rightarrow n=167 \\
<br/> <br/>\therefore \mathrm{S}=\frac{n}{2}[2 a+(n-1) d] \\
<br/> <br/>=\frac{167}{2}[2 \times 3+(167-1) \times 6] \\
<br/> <br/>=\frac{167}{2}[1002]=167 \times 501=83667
<br/> <br/>\end{array}$</div>


title: the-sum-of-all-the-solution-of-the-equation-cos-cos-3-cos-3-4-1-0-6
slug: the-sum-of-all-the-solution-of-the-equation-cos-cos-3-cos-3-4-1-0-6-95716
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
question:
<div class="question">The sum of all the solution of the equation $\cos \theta \cos \left(\frac{\pi}{3}+\theta\right) \cos \left(\frac{\pi}{3}-\theta\right)=\frac{1}{4}, \theta \in[0,6 \pi]$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$15 \pi$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$30 \pi$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{100 \pi}{3}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$30 \pi$
<br/></span> </div>

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp; 2 \cos \theta\left[\cos 120^{\circ}+\cos 2 \theta\right]=1 \\
<br/>\Rightarrow &amp; 2 \cos \theta\left(-\frac{1}{2}+2 \cos ^2 \theta-1\right)=1 \\
<br/>\Rightarrow &amp; 4 \cos ^3 \theta-3 \cos \theta-1=0 \\
<br/>\Rightarrow &amp; \cos 3 \theta=1=\cos 0 \\
<br/>\Rightarrow &amp; 3 \theta=2 n \pi \text { or } \theta=\frac{2 n \pi}{3}, n \in z
<br/>\end{aligned}
<br/>$$
<br/>Given the values so that $2 n$ does not exceed 18 .
<br/>Hence, the sum $=\frac{2 \pi}{3} \sum_1^9 n=\frac{2 \pi}{3} \times \frac{9(9+1)}{2}=30 \pi$</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
title:
the-sum-of-mean-and-variance-of-a-binomial-distribution-b-d-for-3-trials-is-2-7-then-the-binomial-distribution-b-d-is-given-by

slug:
the-sum-of-mean-and-variance-of-a-binomial-distribution-b-d-for-3-trials-is-2-7-then-the-binomial-distribution-b-d-is-given-by-49727

question:
<div class="question">The sum of mean and variance of a Binomial distribution <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> trials is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>7</mn></math>, then the binomial distribution <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>B</mi><mi>D</mi></mrow><mo>→</mo></mover></math> is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>2</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>8</mn></mrow></mfenced><mn>5</mn></msup></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>3</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>7</mn></mrow></mfenced><mn>5</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>6</mn></mrow></mfenced><mn>5</mn></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>3</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>7</mn></mrow></mfenced><mn>5</mn></msup></math></span> </div>

answer:
<div class="solution"><p>Mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>m</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>D</mi><mo>=</mo><mi>n</mi><mi>p</mi></math></p><p>and variance <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msup><mi>σ</mi><mn>2</mn></msup></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>D</mi><mo>=</mo><mi>n</mi><mi>p</mi><mi>q</mi></math></p><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>p</mi><mo>+</mo><mi>n</mi><mi>p</mi><mi>q</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>7</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>n</mi><mi>p</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>q</mi></mrow></mfenced><mo>=</mo><mn>2</mn><mo>.</mo><mn>7</mn><mo>⇒</mo><mi>p</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>q</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mo>.</mo><mn>7</mn></mrow><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>p</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>q</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>9</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi>q</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>q</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>9</mn></math>  [<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mn>1</mn></math>]</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>1</mn><mo>-</mo><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>.</mo><mn>9</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>⇒</mo><mi>q</mi><mo>=</mo><mo>±</mo><mn>0</mn><mo>.</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math>  [<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><mo> </mo><mi>q</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn></math> can't possible as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>q</mi><mo>≤</mo><mn>1</mn></math>]</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>p</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>7</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>B</mi><mi>D</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>3</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>7</mn></mrow></mfenced><mn>5</mn></msup></math></p></div>


title: the-sum-of-the-infinite-series-2-2-2-4-2-4-6-2-6-is-equal-to
slug: the-sum-of-the-infinite-series-2-2-2-4-2-4-6-2-6-is-equal-to-66260
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
question:
<div class="question">The sum of the infinite series $\frac{2^{2}}{2 !}+\frac{2^{4}}{4 !}+\frac{2^{6}}{6 !}+\ldots$
<br/> <br/>is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{e}^{2}+1}{2 \mathrm{e}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{e}^{4}+1}{2 \mathrm{e}^{2}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\left(\mathrm{e}^{2}-1\right)^{2}}{2 \mathrm{e}^{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\left(\mathrm{e}^{2}+1\right)^{2}}{2 \mathrm{e}^{2}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\left(\mathrm{e}^{2}-1\right)^{2}}{2 \mathrm{e}^{2}}$</span> </div>

answer:
<div class="solution">We know that
<br/> <br/>$\frac{e^{x}+e^{-x}}{2}=1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\frac{x^{6}}{6 !}+\ldots$
<br/> <br/>keeping $\mathrm{x}=2$, we get
<br/> <br/>Expression $=\frac{1}{2}\left[\frac{e^{2}+e^{-2}}{2}\right]-1=\frac{\left(e^{2}-1\right)^{2}}{2 e^{2}}$</div>


title: the-sum-of-the-series-1-22-3-2-2-4-2-3-100-2-99-is
slug: the-sum-of-the-series-1-22-3-2-2-4-2-3-100-2-99-is-82401
question: <div class="question">The sum of the series $1+2.2+3.2^{2}+4.2^{3}+\ldots .+100.2^{99}$ is</div>
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100 \cdot 2^{100}+1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$99.2^{100}+1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$99.2^{100}-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$100.2^{100}+1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$99.2^{100}+1$</span> </div>

answer:
<div class="solution">Let $S=1+2.2+3.2^{2}+4.2^{2}+\ldots .+100.2^{99}$
<br/> <br/>It is an arithmetico - geometric series. On multiplying Eq.
<br/> <br/>(i) by 2 and then subtracting it from Eq. (i), we get
<br/> <br/>$S=1+2.2+3.2^{2}+4.2^{3}+\ldots+100.2^{99}$
<br/> <br/>$2 S=1.2+2.2^{2} \pm 3.2^{3}+\ldots+99.2^{99}+100.2^{100}$
<br/> <br/>$\begin{aligned}-S &amp;=1+2+2^{2}+2^{3} \ldots+2^{99}-100.2^{100} \\ &amp; \Rightarrow-S=\frac{1\left(2^{100}-1\right)}{2-1}-100.2^{100} \\ &amp; \Rightarrow-S=2^{100}-1-100.2^{100} \\ &amp; \Rightarrow-S=-1-99.2^{100} \\ &amp; \Rightarrow S=99.2^{100}+1 \end{aligned}$</div>


title: the-sum-to-infinite-term-of-the-series-1-3-2-3-2-6-3-3-10-3-4-14-is-1
slug: the-sum-to-infinite-term-of-the-series-1-3-2-3-2-6-3-3-10-3-4-14-is-1-88125
tags: ['Mathematics', 'Sequences and Series', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>3</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The sum to infinite term of the series $1+\frac{2}{3}+\frac{6}{3^2}+\frac{10}{3^3}+\frac{14}{3^4}+\cdots$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>6</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>2</span> </li> </ul>

answer:
<div class="solution">We have<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/v4Z1kTVNykWyCrgSFIT-Hx_GqWd1cn_W85BMAa4x06o.original.fullsize.png"/><br/><br/>Multiplying both sides by $\frac{1}{3}$, we get<img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/rGNr5MT4r0M2gB_qD0WW5Cd7tquZXrpCI5abkeFzpcU.original.fullsize.png"/><br/><br/>Subtracting eqn. (ii) from eqn. (i), we get<br/>$\begin{aligned} &amp; \frac{2}{3} S=1+\frac{1}{3}+\frac{4}{3^2}+\frac{4}{3^3}+\frac{4}{3^4}+\ldots \ldots \ldots \infty \\ &amp; \Rightarrow \frac{2}{3} S=\frac{4}{3}+\frac{4}{3^2}+\frac{4}{3^3}+\frac{4}{3^4}+\ldots \ldots \ldots \infty \\ &amp; \Rightarrow \frac{2}{3} S=\frac{\frac{4}{3}}{1-\frac{1}{3}}=\frac{4}{3} \times \frac{3}{2} \Rightarrow S=3\end{aligned}$</div>


title: the-sum-to-n-terms-of-the-series-2-1-4-3-8-7-16-15-is
slug: the-sum-to-n-terms-of-the-series-2-1-4-3-8-7-16-15-is-82987
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P3.db
question:
<div class="question">The sum to $\mathrm{n}$ terms of the series
<br/> <br/>$\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots \ldots \ldots \ldots$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$n-1-2^{-n}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$n-1+2^{-n}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1+2^{-n}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$n-1+2^{-n}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots \\=&amp;\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{8}\right)+\left(1-\frac{1}{16}\right)+\ldots . \\=&amp; \mathrm{n}-\frac{\frac{1}{2}\left\{1-\frac{1}{2^{\mathrm{n}}}\right\}}{1-\frac{1}{2}}=\mathrm{n}-1+2^{-\mathrm{n}} \end{aligned}$</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P3.db
title:
the-surface-charge-density-of-a-thin-charged-disc-of-radius-r-is-the-value-of-the-electric-field-at-the-centre-of-the-disc-is-2-0-with-respect-to-the--1

slug:
the-surface-charge-density-of-a-thin-charged-disc-of-radius-r-is-the-value-of-the-electric-field-at-the-centre-of-the-disc-is-2-0-with-respect-to-the-1-99969

question:
<div class="question">The surface charge density of a thin charged disc of radius $\mathrm{R}$ is $\sigma$. The value of the electric field at the centre of the disc is $\frac{\sigma}{2 \epsilon_{0}}$. With respect to the field at the centre, the electric field along the axis at a distance $\mathrm{R}$ from the centre of the $\operatorname{disc}$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">reduces by $70.7 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">reduces by $29.3 \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">reduces by $9.7 \%$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">reduces by $14.6 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">reduces by $70.7 \%$</span> </div>

answer:
<div class="solution">Electric field intensity at the centre of the disc.
<br/> <br/>\(\mathrm{E}=\frac{\sigma}{2 \epsilon_{0}} \text { (given) }\)
<br/> <br/>Electric field along the axis at any distance \(x\) from the centre of the disc
<br/> <br/>\(\mathrm{E}^{\prime}=\frac{\sigma}{2 \epsilon_{0}}\left(1-\frac{\mathrm{x}}{\sqrt{\mathrm{x}^{2}}-\mathrm{R}^{2}}\right)\)
<br/> <br/>From question, \(\mathrm{x}=\mathrm{R}\) (radius of disc)
<br/> <br/>\(\begin{aligned}
<br/> <br/>&amp;\therefore E^{\prime}=\frac{\sigma}{2 \epsilon_{0}}\left(1-\frac{R}{\sqrt{R^{2}+R^{2}}}\right) \\
<br/> <br/>&amp;=\frac{\sigma}{2 \epsilon_{0}}\left(\frac{\sqrt{2} R-R}{\sqrt{2} R}\right) \\
<br/> <br/>&amp;=\frac{4}{14} E
<br/> <br/>\end{aligned}\)
<br/> <br/>\(\therefore\) \% reduction in the value of electric field
<br/> <br/>\(=\frac{\left(\mathrm{E}-\frac{4}{14} \mathrm{E}\right) \times 100}{\mathrm{E}}=\frac{1000}{14} \%=70.7 \%\)</div>


tags: ['Physics', 'Magnetic Properties of Matter', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
the-susceptibility-of-a-magnetism-at-300-k-is-12-1-0-5-the-temperature-at-which-the-susceptibility-increases-to-18-1-0-5-is

slug:
the-susceptibility-of-a-magnetism-at-300-k-is-12-1-0-5-the-temperature-at-which-the-susceptibility-increases-to-18-1-0-5-is-83618

question:
<div class="question">The susceptibility of a magnetism at $300 \mathrm{~K}$ is $1.2 \times 10^{-5}$. The temperature at which the susceptibility increases to $1.8 \times 10^{-5}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$150 \mathrm{~K}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$200 \mathrm{~K}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$250 \mathrm{~K}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$20 \mathrm{~K}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$200 \mathrm{~K}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \chi=\frac{\mathrm{C}}{\mathrm{T}} \\ &amp; \Rightarrow \frac{\chi_{1}}{\chi_{2}}=\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}} \Rightarrow \frac{1.2 \times 10^{-5}}{1.8 \times 10^{-5}}=\frac{\mathrm{T}_{2}}{300} \\ &amp; \Rightarrow \mathrm{T}_{2} \Rightarrow \frac{12}{18} \times 300=200 \mathrm{~K} \end{aligned}$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
title:
the-temperature-of-an-ideal-gas-is-increased-from-2-7-c-to-12-7-c-then-percentage-increase-in-v-rms-is

slug:
the-temperature-of-an-ideal-gas-is-increased-from-2-7-c-to-12-7-c-then-percentage-increase-in-v-rms-is-11203

question:
<div class="question">The temperature of an ideal gas is increased from $27^{\circ} \mathrm{C}$ to $127^{\circ} \mathrm{C}$, then percentage increase in $\mathrm{v}_{\mathrm{rms}}$
<br/> <br/>is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$37 \%$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$11 \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$33 \%$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$15.5 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$15.5 \%$</span> </div>

answer:
<div class="solution">We know, $\mathrm{V}_{\mathrm{mms}}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}}$
<br/> <br/>$\Rightarrow \%$ increase in
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\mathrm{V}_{\mathrm{rms}}=\frac{\sqrt{\frac{3 \mathrm{R} \mathrm{T}_{2}}{\mathrm{M}}}-\sqrt{\frac{3 \mathrm{RT}_{1}}{\mathrm{M}}}}{\sqrt{\frac{3 \mathrm{RT}_{1}}{\mathrm{M}}}} \times 100 \\
<br/> <br/>=\frac{\sqrt{\mathrm{T}_{2}}-\sqrt{\mathrm{T}_{1}}}{\sqrt{\mathrm{T}_{1}}} \times 100 \\
<br/> <br/>=\frac{\sqrt{400}-\sqrt{300}}{\sqrt{300}} \times 100 \\
<br/> <br/>=\frac{20-17.32}{17.32} \times 100=15.5 \%
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'BITSAT', 'BITSAT 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2 u$</span> </div>
source_db: MarksBatch2_P3.db
title:
the-temperature-of-an-ideal-gas-is-increased-from-120-k-to-480-k-if-at-120-k-the-root-mean-square-velocity-of-the-gas-molecules-is-v-at-480-k-it-becom

slug:
the-temperature-of-an-ideal-gas-is-increased-from-120-k-to-480-k-if-at-120-k-the-root-mean-square-velocity-of-the-gas-molecules-is-v-at-480-k-it-becom-67357

question:
<div class="question">The temperature of an ideal gas is increased from $120 \mathrm{~K}$ to $480 \mathrm{~K}$. If at $120 \mathrm{~K}$, the root mean square velocity of the gas molecules is $v$, at $480 \mathrm{~K}$ it becomes</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4 u$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 u$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{u}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{u}{4}$</span> </li> </ul>

answer:
<div class="solution">\(\frac{\mathrm{v}_{1}}{\mathrm{v}_{2}}=\frac{\sqrt{\frac{3 \mathrm{RT}_{1}}{\mathrm{M}}}}{\sqrt{\frac{3 \mathrm{RT}_{2}}{\mathrm{M}}}}=\sqrt{\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}} \therefore \mathrm{v}_{2}=2 \mathrm{v}_{1}\)</div>


tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
title:
the-term-independent-of-x-in-the-expansion-of-left9-mathrmxfrac13-sqrtmathrmxright18-mathrmx-0-is-a-times-the-corresponding-binomial-coefficient-then-

slug:
the-term-independent-of-x-in-the-expansion-of-left9-mathrmxfrac13-sqrtmathrmxright18-mathrmx-0-is-a-times-the-corresponding-binomial-coefficient-then-58054

question:
<div class="question">The term independent of \(x\) in the expansion of \(\left(9 \mathrm{x}-\frac{1}{3 \sqrt{\mathrm{x}}}\right)^{18}, \mathrm{x} &gt; 0\), is a times the corresponding binomial coefficient. Then a is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(1 / 3\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(-1 / 3\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">\(\begin{aligned}<br/>\mathrm{T}_{\mathrm{r}+1} &amp; ={ }^{18} \mathrm{C}_{\mathrm{r}}(9 \mathrm{x})^{18-\mathrm{r}}\left(-\frac{1}{3 \sqrt{\mathrm{x}}}\right)^{\mathrm{r}} \\<br/>&amp; =(-\mathrm{r})^{\mathrm{r}}{ }^{18} \mathrm{C}_{\mathrm{r}} 9^{18-\frac{3 \mathrm{r}}{2}} \mathrm{x}^{18-\frac{3 \mathrm{r}}{2}}<br/>\end{aligned}\)<br/>is independent of \(\mathrm{x}\) provided \(\mathrm{r}=12\) and then \(\mathrm{a}=1\).</div>


tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
the-threshold-wavelength-of-the-tungsten-is-2300-a-if-ultraviolet-light-of-wavelength-1800-a-is-incident-on-it-then-the-maximum-kinetic-energy-of-phot

slug:
the-threshold-wavelength-of-the-tungsten-is-2300-a-if-ultraviolet-light-of-wavelength-1800-a-is-incident-on-it-then-the-maximum-kinetic-energy-of-phot-56814

question:
<div class="question">The threshold wavelength of the tungsten is 2300 $Ãâ¦ .$ If ultraviolet light of wavelength $1800 Ãâ¦$ is incident on it, then the maximum kinetic energy of photoelectrons would be about -</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1.49 \mathrm{eV}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.2 \mathrm{eV}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3.0 \mathrm{eV}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5.0 \mathrm{eV}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.49 \mathrm{eV}$</span> </div>

answer:
<div class="solution">$\mathrm{K}_{\max }=\mathrm{hv}-\mathrm{hv}_{0}=\mathrm{hc}\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right)$
<br/> <br/>$1.24 \times 10^{-6}\left(\frac{10^{8}}{18}-\frac{10^{8}}{23}\right)=1.49 \mathrm{eV}$</div>


tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
title:
the-time-period-of-a-bob-performing-simple-harmonic-motion-in-water-is-2-s-if-density-of-bob-is-4-3-10-3-kg-m-3-then-time-period-of-bob-performing-sim

slug:
the-time-period-of-a-bob-performing-simple-harmonic-motion-in-water-is-2-s-if-density-of-bob-is-4-3-10-3-kg-m-3-then-time-period-of-bob-performing-sim-98814

question:
<div class="question">The time period of a bob performing simple harmonic motion in water is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">s</mi></math>. If density of bob is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo> </mo><mi>kg</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></math>, then time period of bob performing simple harmonic motion in air will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi mathvariant="normal">s</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi mathvariant="normal">s</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">s</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">s</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">s</mi></math></span> </div>

answer:
<div class="solution"><p>Given, density of bob, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo> </mo><mi>kg</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></math><br/>Density of water, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mo>=</mo><msup><mn>10</mn><mn>3</mn></msup><mo> </mo><mo> </mo><mi>kg</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></math></p><p>If <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>'</mo></msup></math> be gravitational acceleration in water then,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mo>'</mo></msup><mo>=</mo><mi>g</mi><mo> </mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>σ</mi><mi>ρ</mi></mfrac></mrow></mfenced><mo>=</mo><mi>g</mi><mo> </mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><msup><mn>10</mn><mn>3</mn></msup><mrow><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup></mrow></mfrac></mrow></mfenced><mo>=</mo><mfrac><mi>g</mi><mn>4</mn></mfrac></math></p><p>As, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>  </mo><msub><mi>T</mi><mrow><mi>air</mi><mo> </mo></mrow></msub><mo>=</mo><mn>2</mn><mi>π</mi><msqrt><mfrac><mi>I</mi><mi>g</mi></mfrac></msqrt></math>,</p><p>Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>  </mo><msub><mi>T</mi><mrow><mi>water</mi><mo> </mo></mrow></msub><mo>=</mo><mn>2</mn><mi>π</mi><msqrt><mfrac><mi>I</mi><msup><mi>g</mi><mo>'</mo></msup></mfrac></msqrt><mo> </mo><mfenced close="]" open="["><mrow><msub><mi>T</mi><mrow><mi>water</mi><mo> </mo></mrow></msub><mo>=</mo><mn>2</mn><mo> </mo><mi mathvariant="normal">s</mi></mrow></mfenced><mo> </mo><mfenced><mrow><msup><mi>g</mi><mo>'</mo></msup><mo>=</mo><mfrac><mi>g</mi><mn>4</mn></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>=</mo><mn>2</mn><mi>π</mi><mo> </mo><msqrt><mfrac><mi>I</mi><mrow><mi>g</mi><mo>/</mo><mn>4</mn></mrow></mfrac></msqrt><mo>=</mo><mn>2</mn><mi>π</mi><mo> </mo><msqrt><mfrac><mi>I</mi><mi>g</mi></mfrac></msqrt><mo> </mo><mo>·</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>=</mo><mn>2</mn><mo>.</mo><msub><mi>T</mi><mrow><mi>air</mi><mo> </mo></mrow></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo> </mo><msub><mi>T</mi><mrow><mi>air</mi><mo> </mo></mrow></msub><mo>=</mo><mn>1</mn><mo> </mo><mi mathvariant="normal">s</mi></math></p></div>


